(Describer) Titles: Welcome to Algebra 1: Factoring Trinomials with a Leading Coefficient Not Equal to 1.

Hi, guys. Welcome to Algebra 1. Today's lesson will focus on factoring trinomials when your leading coefficient is not 1. What you know about factoring trinomials with a leading coefficient of 1 will help you get through these problems. You ready? Let's go.

(Describer) On screen, x-squared plus 5x plus 6.

Okay, so to jog your memory, let's first think back about how you would factor this trinomial, for example. You'd probably start out by setting up what the answer is going to look like: with your sets of parentheses. You know you'll have some X's out front, like that.

(Describer) She writes on the screen.

Then you need the factors of 6. I know 1 times 6 will give me 6, and so will 2 times 3. And I need the pair of factors that, in this case, will combine to give me 5. So when I combine 1 and 6, I get 7. That's not the pair. 2 and 3, when I combine those, I get 5. That is the pair that I want. I'd say my factors are X plus 2, and X plus 3. Here, we factored a trinomial that had a leading coefficient of 1. There's an invisible 1 there, for that first term. What happens when you have something like this?

(Describer) Three x-squared plus 10x plus 8.

(female narrator) 3X squared plus 10X plus 8. When you have a trinomial and the leading coefficient is not 1? To handle these types of problems, there's a few steps you should learn, then we'll return to that problem. There's a few; there's five. The first thing to remember, looking at the top, is your trinomial is written in the form AX squared plus BX plus C. The first step is multiplying the A and the C. Then figure out the factors of that product that will add up to give us B, that middle number, like you were doing before. Then we'll do something called splitting the middle term. I'll show you what I mean by that. Then we're going to factor our two binomials, and then we'll have our answer. Keep these steps written on the side to follow along while we factor this trinomial. The first thing I do is, I know this trinomial is written in the form AX squared plus BX plus C. You need to first factor the A and the C, or the first number and the last number. You need to multiply those together, I'm sorry. So 3 times 8... that's 24, right? So I've found the product of A and C. Now I need to figure out what factors of 24 would also combine and give me 10? This step we've handled before, as far as listing our factors and figuring out the right sum. In this problem, there's an extra step of multiplying that A and that C together, to start with. Now that we've got 24, let's list our factors. So 1 and 24; 2 and 12; 3 and 8; and 4 and 6. All those pairs will make 24 when you multiply them. Now I need to figure out which pair of factors would give me a sum of 10? Let's start testing these out. Combining 1 and 24, that's 25. 2 and 12, that's 14. 3 and 8, that's 11. 4 and 6, that is 10, right? So these 3 didn't work, but the 4 and the 6 did work. Now what I'm going to do, I'm going to bring this off to the side of it. Let's erase this. I'm at the point now where I split the middle term. What that means is that instead of writing-- this is my middle term, that 10X. I'm going to split it, so instead of writing 10X, I'm going to write it as a sum of two other terms. I'll split it into two terms. So I'll still have the 3X squared, but instead of writing 10X, I'm going to write 4X plus 6X-- going back over here to our factors-- because 4 and 6, I know will add to give me 10. So 4X plus 6X gives me 10X. You're not changing the meaning of the problem, you're just representing that 10X differently. Not 64X, because that would change the meaning of your problem-- it's 6X. There we go, plus 8. So like I said, 10X, we're just representing it as 4X plus 6X, because that is still 10X. We're doing this just to get through this problem. Let me get this out of our way. Now, what we do is we look at this like it's two different binomials. Like we have this first group, right here, and then we have the second group, right here. And we'll factor this first group, and then we'll factor the second group. Remember, when we factored binomials, we were looking for our greatest common factor, right? Between 3X squared and 4X. First, just consider your 3 and your 4. There's no factors that 3 and 4 have in common, except 1, right? So I can just keep that in my mind. Now, 8 squared and X-- you may need to see this. Let's go ahead and work to the side. We may soon not need to write it. So factor in that 3-- 1 and 3, I know, right? It's prime, so that's it for 3. Then factoring X squared, you know you have X and X, and that's it for 3X squared. Now, factoring 4X, I know 1 times 4 would give me 4, and so would 2 times 2. And then I do have one factor of X that I can write. When I need the greatest common factor-- in this case, all they have in common is a 1-- that doesn't change anything, really--and the X. I'll say, all right, the greatest common factor of those two terms is X. I'll write my X, open up my parentheses, then I need to divide each of those terms by X. So 3X squared divided by X. That's like an invisible 1 I don't see. So that's like 3 divided by 1, which is 3. Then X squared divided by X to the 1st, so subtracting your exponents, that's just X to the first. So I know, right in here, I'd have a 3X. Let me go ahead and write that plus.

(Describer) ...in the first parentheses.

Now I need to divide 4X by X. So 4X divided by X, or like a 1X. 4 divided by 1, that's just 4. X to the 1st divided by X to the 1st, subtracting those exponents, that's X to the 0, and anything to the 0 power is just 1. This is really just like 4 times 1, which is just 4. That means the second group-- or the second spot-- would have a 4.

(Describer) 3x plus 4.

We do the same process with that 6X plus 8. Okay, let's get this work out of our way. We need to find the greatest common factor between 6X and 8. So we have a 6X, we have an 8. So as far as the 6 is concerned, I know 1 times 6 would work, and so would 2 times 3. Then I have a factor of X. As far as my 8 is concerned, I know 1 times 8 would work, and so would 2 times 4. In this case, my greatest common factor is 2. Greatest common factor is 2. Ooh--slid that down by accident. So what I would write here now is plus 2, because that's the greatest common factor for the second group, right here. Open up my parentheses. Go ahead and throw that plus sign in there. Now I'll divide each of these terms by the greatest common factor. I'm ignoring this work over here for a minute. So 6X divided by 2. 6 divided by 2, that's 3. I have an X up top but nothing on the bottom, so it's just carried along. So I've got 3X for the first term. Now I need to divide 8 by 2. Well, 8 divided by 2, that's 4. Then you see in your second spot that you have a 4, then you look. You have the same binomial in your parentheses. So that itself is like a common factor of this trinomial you were trying to factor to start with. To write your final answer, this is what you do. You have that X and this 2 that are the greatest common factors. When you factor those groups, you bring them together. You'd have X plus 2 as one factor. Then that 3X plus 4 that's common to both. That's another factor. And you are all done. You have factored that trinomial that had a leading coefficient that was not 1. There's a lot to these types of problems. We definitely have to do a few so you can get the hang of it. Let's try another one. Now I'm factoring 3X squared plus 7X plus 2. You remember the first step. You multiply the A and the C. In this case, the 3 and the 2. So 3 times 2... I know that's 6. Now I need factors of 6 that are going to combine to give me 7, okay? So I start listing: 1 times 6; 2 times 3. I need a positive 7. So 1 and 6, that's 7. You got it, it's that first pair. This is the point where we split that middle term, so I'll rewrite this trinomial as 3X squared plus X... plus 6X plus 2. I'm splitting that 7X into 1X plus 6X. It doesn't change the meaning at all; it just helps factor this trinomial a little better. Now that I've split the middle term, I'm at the point now where I need to factor these two groups, these two pairs of binomials. I'm going to get this work out of our way. Let's consider... this first group and then the second group. Let's see how we're doing with finding greatest common factors here. Between these two terms, the greatest common factor. The coefficient of this one's 3 and of this one is 1. As far as my numbers are concerned, the greatest number they have in common as a factor is 1. Now I see I have X squared and I have X, so if this term has 2X's and this term has 1X, they both have at least 1X in common. So the greatest common factor of these 2 terms is just 1X.

(Describer) She writes X.

I'm going to open up the parentheses. And now it's time to divide each of these terms by X. We'll come off over here. So 3X squared divided by X, which is like a 1X, really, to the 1st because we don't see that exponent. So 3 divided by 1, that's 3. X squared divided by X to the 1st, that's just X because we subtract our exponents here. For my first term in here, I know I have a 3X; it'll be plus, so get that in there. For the second term, I need to divide X by X. Well, X divided by X, that's just 1. That means I have a 1 in here, for that second group of parentheses, or that second term in the parentheses. It's time to do the same thing with 6X plus 2. Let me get this work out of our way.

(Describer) So far, the answer is x times 3x plus 1.

All righty. I need the greatest common factor between 6X and 2. I see that that's just a 2; it's a constant term. They won't have any X's in common. I need to focus on these numbers here. As far as 6 goes and 2 goes-- 2 is prime, so the only factors for 2 are 1 and 2. And 2 is also a factor of 6, so between these two terms, my greatest common factor is going to be 2. So I'll write plus 2. Go ahead and open my parentheses. I know I'm going to have a plus sign in between. Now it's time to divide each of these terms by our greatest common factor, over here. So 6X divided by 2. So 6 divided by 2, that's 3. This X is just hanging along for the ride, so 3X. Then I have 2 divided by 2... which is 1. So then you look. Here you notice you have the same group in those parentheses, so we're at the point of rewriting this. Here we have our X plus 2, which is out front. We bring that together, so X plus 2. And then that factor that was repeated: 3X plus 1. And you are all done. You factored that trinomial. There are a few steps to it, but do it a few times and you get the hang of it. Let's try some more. Factor completely: 2X squared plus 5X plus 3. Again, we need to start out by multiplying 2 times 3. 6 just keeps popping up here. So 2 times 3 is 6. So factors of 6: 1 and 6... 2 and 3. Now we need the factors of 6 that are going to combine to give us 5. Let's see what we got here. 1 and 6, that's 7. 2 and 3, that is 5. So I'll split this middle term and rewrite this as 2X squared plus 2X plus 3X plus 3. Okay? Do you remember what we do next? We're at the point where we take this first group, we take the second group, and factor each group. We pull out the greatest common factor in each group. Let's get this work out of the way. So 2X squared and 2X. They both have that 2 in common. The first term has X squared, so there's a product of two X's with that one. The second term is 2X-- one X for that one. For the numbers, 2 was the greatest common factor. For the variables, X was the greatest common factor. So my greatest common factor: 2X. Open up my parentheses. I know I have a plus in between. Now I'm going to divide each of these terms by that greatest common factor. So 2X squared divided by 2X. So 2 divided by 2, that's 1. X squared divided by X to the 1st, that's X, so basically it's X. When I divide over there, I get X. Then I have 2X divided by 2X. Well, anything divided by itself is 1, so we know that 2X divided by 2X is 1. We've handled that first set. Let's go on to the next group, this 3X plus 3. Let me get this work out of the way. All righty. So 3X and a 3. They both have 3-- there's a 3 involved in each term. For my number, the greatest common factor is 3. The first term has an X, the second term doesn't, so they don't have an X in common. They only have a 3 in common. I'll write my greatest common factor, 3, for this group, open up my parentheses, go ahead and put the plus inside, and start dividing. So 3X divided by 3. Well, 3 divided by 3, I know that's 1. X on top and no X on the bottom, so that carries along for the ride. I have, basically, 1X, or just X. Now, 3 divided by 3... that's 1, so you got a 1.

(Describer) 2x times, open parentheses, x plus 1, close parentheses, plus 3 times, open parentheses, x plus 1, close parentheses.

(female narrator) 2X times open parentheses, X plus 1, close parentheses, plus 3 times, open parentheses, X plus 1, close parentheses. All right, remember what we do next? We're at the point now to clean it up and write our final answer. We've got the 2X out here, the 3 out front here. In one set of parentheses you'd have 2X plus 3.

(Describer) The first factor in the answer.

Then you'd have that factor that was in both of those-- the repeated one, that X plus 1.

(Describer) The other factor.

And you are all done, you've factored completely. Okay? Let's try another.

(female narrator) 4X squared minus X minus 3.

(Describer) 4x-squared minus x minus 3.

Okay, so this one's involving some negatives. That'll be a little different. Other than that, the process is exactly the same. Start out by multiplying A and C. So 4 times 3, that's 12. And I know I need factors of 12 that will combine to give me negative 1. Once I list these factors, they'll have different signs: One number must be positive and the other negative, if when I combine them, I get negative 1. Let's list the factors: 1 and 12; 2 and 6; 3 and 4, right? Now we have to find which one of these pairs of numbers will work and which should be negative. We know we're trying to get negative 1, so throw the first pair out. There's no way to combine a 1 and a 12 and get negative 1, no matter which were negative. Negative 1 plus 12 would be 11. 1 plus negative 12 would be negative 11. So there's no way it's the first pair. 2 and 6? If the 2 were negative, it'd be negative 2 plus 6, and that's 4. If the 6 were negative, then it'd be a negative 4. So that's not it; it has to be this last pair. I need to find which number needs the negative. Negative 3 plus 4 would be positive 1. And I needed a negative 1, so that means it's the 4 that's negative. 3 plus negative 4 will give me negative 1. Now I'm ready to go ahead and split my middle term. I'm going to rewrite it as-- I've got my 4X squared-- I'm going to write plus 3X minus 4X minus 3, okay? Do you remember what comes next? Go ahead, you take your groups... and you start factoring the pairs, okay? Let's get some more room to work up here. If I look at these first terms, the 4X squared and the 3X. 4 and 3 have nothing in common but 1, but I do see I have an X squared and an X. These terms each have an X as a factor. So I'm going to write X as the greatest common factor, okay? Let's get the parentheses up, and get the plus sign in there. So 4X squared divided by X-- I'm ready to start dividing these terms by my greatest common factor. That's essentially a 1X to the 1st over there. So 4 divided by 1. That's 4X squared divided by X, that's just X. So now I have 4X over here. And then 3X divided by X. So 3X to the 1st divided by 1X to the 1st. So 3 divided by 1, that's just 3. X to the 1st divided by X to the 1st is X to the 0 power, which is just 1. Basically, 3 times 1, that's not changing anything, so I'll just leave it as 3. Bring it over here. Now I'm at the point to go ahead and look at these factors. I have a negative 4X and a negative 3. The only thing these two terms have in common is a negative 1, that's it. Both of these terms are negative. We knew 3 and 4 had no factors in common. And this is the only term that has the X. The only thing they have in common is a negative 1. Now it's time to go through. Just go ahead and get that in there.

(Describer) The second set of parentheses.

And let's get this work out of the way. Here we go. Now, I'm not going to write the sign in here just yet because I want to show you something over here. Now, I have negative 4X divided by negative 1. So negative 4X divided by negative 1. Negative 4 divided by negative 1 is just 4. X on top, but no X on the bottom, so it carries along for the ride. So I get 4X over here. Then I have a negative 3 divided by negative 1. So negative 3 divided by negative 1, that's positive 3. So plus 3, okay?

(Describer) It's another 4x plus 3.

Remember what comes next? You're at the finish line, just about. It's time to just write that final answer. So I see I have the X out front here and the negative 1 out front here. So X minus 1.

(Describer) The first factor.

And both of these have the 4X plus 3. So 4X plus 3, and you're all done.

(Describer) The second factor.

Exactly like we saw when we were factoring trinomials where the leading coefficient was 1, check your answer by multiplying your factors out. At the last step, you write your factors. To check your answer, multiply your factors together. Use FOIL and multiply them together to get the starting problem. You can still check your answers that way.

(Describer) Next, 2x-squared plus x minus 15.

Okay? Let's continue. Got one more here. So you remember the first step? Good. Multiply A and C together. So 2 times 15--because I'm going to ignore the sign for a second--is 30. So I'm going to factor 30. So 1 times 30; 2 times 15; 3 times 10; and 5 times 6-- and that's it for 30. I need the factors that will combine to give me a positive 1. And remember, if you consider the sign now, this was really a positive 2 times a negative 30. So that means that-- write the negative because now we're considering the signs-- if the product is negative, then I had to multiply together a positive with a negative number. We need to check which pair is going to work, where one number is positive and one is negative, and the answer's 1 for that middle term. We know there's no way to combine 1 and 30 and the answer be positive 1; there's no way. If the 1 were negative, it'd be 29. And If the 30 were negative, it'd be negative 29. So that's not it. 2 and 15-- if the 2 were negative, the answer would be 13. If the 15 were negative, the answer is negative 13. So that's not it. 3 and 10--if the 3 were negative, it'd be 7. If the 10 were negative, it'd be negative 7, so nope. We know it has to be this pair, but which one's negative? If the 5's negative-- negative 5 plus 6-- that is positive 1. That's the pair that we want: negative 5 and positive 6. Let's split our middle term. We're going to rewrite this as 2X squared minus 5X plus 6X... minus 15, okay? We didn't change the meaning at all; we're just rewriting that 1X as negative 5X plus 6X. Let's get this work out of the way because now it is time to take these two groups and start looking at these separately to get the greatest common factors. If I consider 2X squared and 5X-- 2 and 5 are both prime numbers, so the only factor they have in common is 1. As far as the variable part goes, X squared and an X, they both have at least one factor of X. So the greatest common factor between these two terms is X. Go ahead and open up my parentheses. I'll start dividing here. And I'm going to write my "minus" in this time. So 2X squared divided by X. 2X squared divided by X. There's a 1 and 1 I don't see. So this will end up being: 2 divided by 1 is 2, X squared divided by X is X. So 2X for the first term. And now I have--I've already written my negative over here, but to show you why I did, because I'm dividing negative 5 by X. I knew the answer would be negative, dividing a negative and a positive. Go ahead and put the invisible 1's in. Negative 5 divided by 1 is negative 5. And X divided by X is 1. So basically, my answer's negative 5. And I already wrote the negative part. I'll drop the 5 in there.

(Describer) X times 2x minus 5.

Repeat the same process with that second group. Get that out of the way. Okay, so 6X and 15. You may need to write the factors down for this one. It may not be clear what it is at first. For 6, you have 1 and 6; 2 and 3. You have an X in there, so there's a factor of X. And for 15: 1 and 15; 3 and 5. And now you can look and see. 15 has no X, so there's no factor of X in that one. My greatest common factor here is 3. I'll come back to my problem. I'll put my 3 in there. I'll put the minus sign in there this time because I know, like I did over here, I knew the answer would be a negative. Let's start dividing. Let's get some more room. So 6X divided by 3. That's 2X. 6 divided by 3 is 2. I have an X up top but nothing on the bottom, so it carries along. Then I have negative 15 divided by 3... which is negative 5. And I already wrote the negative part in there. I knew it was going to be negative, so there's my 5. You remember what's next? You're ready to write that final answer. Look and see which terms were out front. You've got the X and the 3. So X plus 3. Then in my parentheses I have 2X minus 5, and 2X minus 5. That means that's that second factor here: 2X minus 5. And you are all done. So that initial trinomial that we started with factors to X plus 3 times 2X minus 5, okay? You're ready to try a couple of these. Go ahead and take some time, remember those steps, and work your way through these. Check your answer before you press play. Press pause and see how you do. To compare with me, press play.

(Describer) Title: Factor completely. Number One: 5x-squared plus 11x plus 2. Number Two: 3x-squared minus 13x plus 4.

(female narrator) Factor completely. Number 1: 5X squared plus 11X plus 2. Number 2: 3X squared minus 13X plus 4. You ready to check? Let's see. When I factored that first trinomial, I got 5X plus 1, and X plus 2. And when I factored the second one, I got 3X minus 1, and X minus 4. This is how I did it.

(Describer) She goes to the first problem by itself.

First thing, remember, we're following our steps. We multiply the A and the C. So 5 times 2, that's 10. Now I need factors of 10. I need to find the pair of factors that, when I combine them, I get 11. As far as 10 goes, I know 1 times 10 will give me 10. So will 2 times 5. I need the pair that gives me 11. You probably spotted that pretty quickly--the 1 and 10. 1 plus 10 is 11. You know that's the pair of factors you want. Now jump back to your trinomial, and you can split that middle term. So 5X squared plus X plus 10X plus 2. Now you're ready to take it in groups, and find the greatest common factor in each group. As far as 5X squared and 5 goes, the only factor they have in common is X.

(Describer) She writes x and the first parenthesis.

If I go ahead and do my division, like I need to... So 5X squared divided by X... which is basically a 1X to the first, right? 5 divided by 1, that's 5. X squared divided by X to the 1st, that's X. I knew I had 5X right here. Let's close that off, put my plus sign in there. Now I need X divided by X. Well, anything divided by itself is 1. That's how I knew the second term in there was 1. So let's get that work out of the way. Do that second group. Okay. So between 10X and 2. 2's prime, so the only factors are 1 and itself-- 1 and 2. It doesn't have an X, so the greatest common factor between 10X and 2 is 2. I'll write my plus 2, open up my parentheses. I have a plus sign in between. I'm ready to divide to see what actually goes in this group of parentheses. So 10X divided by 2. 10 divided by 2 is 5, X on top but nothing on the bottom... Moving the screen. Let's get that crazy mark out of the way. There we go. Okay, so 10X divided by 2. 10 divided by 2 is 5. X up top but nothing on the bottom, so we're just carrying that along. So in my group, first term is 5X. Now I need to divide 2 by 2. So 2 divided by 2, that's 1. And you got your 5X plus 1. You probably noticed in these problems, if you don't have the same thing in the parentheses, go back and check because something went wrong. These kinds of problems should end up with that same group inside your parentheses. Okay? That's a hint while you're working. Now you're at the point, just bring everything together. You've got your X and you've got that 2 out front. So X plus 2. And then I have my 5X plus 1. And you're all done. If you wrote these in another order, like "5X plus 1" first and "X plus 2" second-- that's fine. You can multiply in any order you want, according to the commutative property. If your factors are reversed from mine, you're still right. Here's how I got that last one. Same process, multiply the A and C. So 3 times 4, that's 12. So I need my factors for 12. So 1 and 12; 2 and 6; 3 and 4. I ignore the signs to start with-- when everything's positive, you can ignore the signs because they'll be positive. It's when there's a negative in there that you want to go, "Something else is going to happen." Ignore them to start, but then consider that it's negative as you work through the problem. I'm there now, where I must consider this middle term's negative. If I'm multiplying to get a positive number, because I know that's positive-- even though it's the 4, and to work through the problem, it's 12, it's still positive; that sign is the same. And the middle term is negative. So the only way I can multiply two numbers together and get something positive, but combine them and get something negative, is if the numbers themselves are both negative. I must consider that these are negative numbers, and then I need to find the pair that will give me negative 13. Negative 1 plus negative 12 is negative 13. What do you know? It was the first one. So that's the pair we need. Now we we can go back to our problem and split that middle term. So 3X squared minus X minus 12X plus 4, okay? I'm going to get that old work out of the way because more is coming. All right, you remember what we do now? Break it up into those groups. So I've got 3X squared minus X. So between these two terms, your greatest common factor is just X. So divide each of these terms by X. So 3X squared divided by X, which is basically a 1X to the 1st, right? So 3 divided by 1, that's 3. X squared divided by X is X. So over here, I've got my 3X. I won't write the minus yet; I'll bring it over here with me and put it with my answer. I'm dividing negative X by positive X. Basically, a negative 1X to the 1st divided by a positive 1X to the 1st. Negative 1 divided by positive 1 is negative 1. X to the 1st divided by X to the 1st. That's X to the 0, which is really 1, which doesn't change anything. If I multiply negative 1 times 1, it's still negative 1. So I'll leave that and bring it over here. Got my minus 1. Now I'm at the point where I can handle the second group. So let's get this old work out of the way. Now I see I have a negative 12X and a positive 4. I need to find out what's the greatest common factor between these two terms? Because this is negative, I'm going to go ahead and pull that out front. I'm going to pull a negative common factor out of each of these-- this is what I mean. As far as 12 goes, I've got 1 and 12; 2 and 6; 3 and 4. As far as 4 goes, I've got 1-- keep sliding that down-- as far as 4 goes, I've got 1 and 4; 2 and 2. So the greatest common factor between these numbers is 4. But I could say that it's negative 4, okay? Because if I treat this like it's a negative 12, which basically it is, then I can say, I already know 4 is my common factor. I'll jump down to and 3 and 4 and say, that's the same as 3 times negative 4, which is negative 12. And negative 1 times negative 4 is also positive 4. So I'll pull out a negative 4 over here for my greatest common factor. A little math magic. Now I'll bring these here and now it's time to divide.

(Describer) ...parentheses.

I need to divide each of those terms by that negative 4. Let's get this work out of the way. So I have negative 12X divided by negative 4... which--ooh, 12X. So negative 12 divided by negative 4, that's 3. X up top but nothing on the bottom, so it just carries along. You see you've got your 3X. Then I have positive 4 divided by negative 4. Positive 4 divided by negative 4 is negative 1. You see we have our minus 1. Now we're at the last step, to go ahead and write the final answer. So I'll bring these together in a group of parentheses. I'd have X minus 4. Then for my second set, 3X minus 1, that factor that was in both of those. Remember, you can always multiply these factors together to check your answer and get that trinomial that we started with. I hope you're feeling confident about factoring trinomials where you leading coefficient is not 1, and that knowing how to factor trinomials where your leading coefficient is 1 helped you through this lesson. Hope to see you back here soon. Bye!

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