Question 119790: factor 32a^2+4a-6
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 
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Looking at we can see that the first term is  and the last term is  where the coefficients are 16 and -3 respectively.
Now multiply the first coefficient 16 and the last coefficient -3 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient 2? Let's list all of the factors of -48:
Factors of -48:
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -48
(1)*(-48)
(2)*(-24)
(3)*(-16)
(4)*(-12)
(6)*(-8)
(-1)*(48)
(-2)*(24)
(-3)*(16)
(-4)*(12)
(-6)*(8)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2
| First Number | Second Number | Sum | | 1 | -48 | 1+(-48)=-47 | | 2 | -24 | 2+(-24)=-22 | | 3 | -16 | 3+(-16)=-13 | | 4 | -12 | 4+(-12)=-8 | | 6 | -8 | 6+(-8)=-2 | | -1 | 48 | -1+48=47 | | -2 | 24 | -2+24=22 | | -3 | 16 | -3+16=13 | | -4 | 12 | -4+12=8 | | -6 | 8 | -6+8=2 |
From this list we can see that -6 and 8 add up to 2 and multiply to -48
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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 Now reintroduce the GCF
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Answer:
So factors to
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