Question 115655: Can someone please help me with this problem. I tried to do the problem and I just keep getting very confused.
We are specifically looking for how to factor polynomials when the leading coefficient is not 1. How can we find the correct way to break up the middle term to make this come out right.
Can you please factor the polynomial
4x^2 + 4x - 99
Please tell us why and how you carry out each step of the process.
PLEASE HELP ME!!!!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 4 and -99 respectively.
Now multiply the first coefficient 4 and the last coefficient -99 to get -396. Now what two numbers multiply to -396 and add to the middle coefficient 4? Let's list all of the factors of -396:
Factors of -396:
1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396
-1,-2,-3,-4,-6,-9,-11,-12,-18,-22,-33,-36,-44,-66,-99,-132,-198,-396 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -396
(1)*(-396)
(2)*(-198)
(3)*(-132)
(4)*(-99)
(6)*(-66)
(9)*(-44)
(11)*(-36)
(12)*(-33)
(18)*(-22)
(-1)*(396)
(-2)*(198)
(-3)*(132)
(-4)*(99)
(-6)*(66)
(-9)*(44)
(-11)*(36)
(-12)*(33)
(-18)*(22)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4
| First Number | Second Number | Sum | | 1 | -396 | 1+(-396)=-395 | | 2 | -198 | 2+(-198)=-196 | | 3 | -132 | 3+(-132)=-129 | | 4 | -99 | 4+(-99)=-95 | | 6 | -66 | 6+(-66)=-60 | | 9 | -44 | 9+(-44)=-35 | | 11 | -36 | 11+(-36)=-25 | | 12 | -33 | 12+(-33)=-21 | | 18 | -22 | 18+(-22)=-4 | | -1 | 396 | -1+396=395 | | -2 | 198 | -2+198=196 | | -3 | 132 | -3+132=129 | | -4 | 99 | -4+99=95 | | -6 | 66 | -6+66=60 | | -9 | 44 | -9+44=35 | | -11 | 36 | -11+36=25 | | -12 | 33 | -12+33=21 | | -18 | 22 | -18+22=4 |
From this list we can see that -18 and 22 add up to 4 and multiply to -396
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 )
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Answer:
So factors to
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