SOLUTION: When factoring the below polynomial I don't understand what happens to 9x+2y-5 going from the first step to the second step. also where did the (-1) and (+5) come from in step 2? i

Question 400567: When factoring the below polynomial I don't understand what happens to 9x+2y-5 going from the first step to the second step. also where did the (-1) and (+5) come from in step 2? i am working from a text book and going from step 1 to step 2 is not explained. the Polynomial is below...
Factor this polynomial...
2x^2+7xy+3y^2+9x+2y-5
(2x+y)(x+3y)+9x+2y-5 step 1
[(2x+y)-1][(x+3y+5] step 2
(2x+y-1)(x+3y+5) step 3 solved
Thank You for your help
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
(2x+y)(x+3y)+9x+2y-5 step 1
To be honest, I've never seen factoring like this before. And there may be a better way of explaining it than what follows.

The key is the -5 at the end. There are only two ways to factor -5: -1 and 5 or 1 and -5. The idea is to insert one of these pairs into the factors at the beginning. The possibilities are:
[(2x+y)-1][(x+3y)+5]
[(2x+y)-5][(x+3y)+1]
[(2x+y)+1][(x+3y)-5]
or
[(2x+y)+5][(x+3y)-1]
Only one of these is correct. To figure out which one you multiply the numbers inserted by the first part of other factor. Then add like terms. Only one of these combinations will result in the "missing" 9x+2y. (The -5 comes from the factors we've inserted.) Let's look at one of the wrong combinations first:
[(2x+y)-5][x+3y]
We will multiply the -5 by the other factor, x+3y and we will also multiply the +1 by the other factor, 2x+y-5
-5(x+3y) = -5x-15y
+1(2x+y) = 2x+y
Adding the like terms we get: -3x-14y not the 9x+2y we're looking for.

The right combination is
[(2x+y)-1][(x+3y)+5]
Hers we will multiply -1 times x+3y and 5 times 2x+y
-1(x+3y) = -x-3y
5(2x+y) = 10x + 5y
Adding the like terms we get 9x+2y!

None of the other combinations give us the 9x+2y we are looking for.

This approach to factoring can be described as reverse FOIL with the "First" parts being (2x+y) and (x+3y).

P.S. If you learn a better way to describe/explain this factoring I would like to hear about it. If you would like to share it with me, you could "thank" me and include this better description/explanation in the note. Thanks.
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