Factor polynomial $$ 108x^{2}+33x-56 $$
$$ 108x^{2}+33x-56 = ( 12x-7 )( 9x+8 ) $$
Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:
Step 2: Multiply the leading coefficient $\color{blue}{ a = 108 }$ by the constant term $\color{blue}{c = -56} $.
$$ a \cdot c = -6048 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -6048 $ and add to $ b = 33 $.
Step 4: All pairs of numbers with a product of $ -6048 $ are:
| PRODUCT = -6048 | |
| -1 6048 | 1 -6048 |
| -2 3024 | 2 -3024 |
| -3 2016 | 3 -2016 |
| -4 1512 | 4 -1512 |
| -6 1008 | 6 -1008 |
| -7 864 | 7 -864 |
| -8 756 | 8 -756 |
| -9 672 | 9 -672 |
| -12 504 | 12 -504 |
| -14 432 | 14 -432 |
| -16 378 | 16 -378 |
| -18 336 | 18 -336 |
| -21 288 | 21 -288 |
| -24 252 | 24 -252 |
| -27 224 | 27 -224 |
| -28 216 | 28 -216 |
| -32 189 | 32 -189 |
| -36 168 | 36 -168 |
| -42 144 | 42 -144 |
| -48 126 | 48 -126 |
| -54 112 | 54 -112 |
| -56 108 | 56 -108 |
| -63 96 | 63 -96 |
| -72 84 | 72 -84 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = 33 }$
| PRODUCT = -6048 and SUM = 33 | |
| -1 6048 | 1 -6048 |
| -2 3024 | 2 -3024 |
| -3 2016 | 3 -2016 |
| -4 1512 | 4 -1512 |
| -6 1008 | 6 -1008 |
| -7 864 | 7 -864 |
| -8 756 | 8 -756 |
| -9 672 | 9 -672 |
| -12 504 | 12 -504 |
| -14 432 | 14 -432 |
| -16 378 | 16 -378 |
| -18 336 | 18 -336 |
| -21 288 | 21 -288 |
| -24 252 | 24 -252 |
| -27 224 | 27 -224 |
| -28 216 | 28 -216 |
| -32 189 | 32 -189 |
| -36 168 | 36 -168 |
| -42 144 | 42 -144 |
| -48 126 | 48 -126 |
| -54 112 | 54 -112 |
| -56 108 | 56 -108 |
| -63 96 | 63 -96 |
| -72 84 | 72 -84 |
Step 6: Replace middle term $ 33 x $ with $ 96x-63x $:
$$ 108x^{2}+33x-56 = 108x^{2}+96x-63x-56 $$Step 7: Apply factoring by grouping. Factor $ 12x $ out of the first two terms and $ -7 $ out of the last two terms.
$$ 108x^{2}+96x-63x-56 = 12x\left(9x+8\right) -7\left(9x+8\right) = \left(12x-7\right) \left(9x+8\right) $$