SOLUTION: Can you help me solve this? Factor completely. 12x^2 - 10x -8 I keep getting confused because I'm supposed to find terms that have a product of negative 8 and a sum of negative

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Question 182121: Can you help me solve this? Factor completely. 12x^2 - 10x -8
I keep getting confused because I'm supposed to find terms that have a product of negative 8 and a sum of negative 10, but all of the answers doesn't factor back out for me.
Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Start with the given expression


Factor out the GCF


Now let's focus on the inner expression




------------------------------------------------------------



Looking at we can see that the first term is and the last term is where the coefficients are 6 and -4 respectively.

Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From this list we can see that 3 and -8 add up to -5 and multiply to -24


Now looking at the expression , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )



------------------------------------------------------------




So our expression goes from and factors further to


------------------
Answer:

So factors to

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
First of all, we factor the trinomial, .
.

We now need to factor the new trinomial
This means that we now have to find factors that have a product of – 24, and a sum of – 5. These factors are – 8 and + 3. Therefore, now becomes: . Now, factor each binomial, beginning with the 1st and then the second (+ 3x – 4).
= 2x(3x – 4) + 1(3x - 4).

Therefore, = (2x + 1)(3x – 4), and
= 2(2x + 1)(3x – 4).
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