Factor polynomial $$ -8x^{2}-13x+21 $$
$$ -8x^{2}-13x+21 = -( x-1 )( 8x+21 ) $$
Step 1 :
After factoring out $ -1 $ we have:
$$ -8x^{2}-13x+21 = - ~ ( 8x^{2}+13x-21 ) $$Step 2 :
Step 2: 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 3: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = -21} $.
$$ a \cdot c = -168 $$Step 4: Find out two numbers that multiply to $ a \cdot c = -168 $ and add to $ b = 13 $.
Step 5: All pairs of numbers with a product of $ -168 $ are:
| PRODUCT = -168 | |
| -1 168 | 1 -168 |
| -2 84 | 2 -84 |
| -3 56 | 3 -56 |
| -4 42 | 4 -42 |
| -6 28 | 6 -28 |
| -7 24 | 7 -24 |
| -8 21 | 8 -21 |
| -12 14 | 12 -14 |
Step 6: Find out which factor pair sums up to $\color{blue}{ b = 13 }$
| PRODUCT = -168 and SUM = 13 | |
| -1 168 | 1 -168 |
| -2 84 | 2 -84 |
| -3 56 | 3 -56 |
| -4 42 | 4 -42 |
| -6 28 | 6 -28 |
| -7 24 | 7 -24 |
| -8 21 | 8 -21 |
| -12 14 | 12 -14 |
Step 7: Replace middle term $ 13 x $ with $ 21x-8x $:
$$ 8x^{2}+13x-21 = 8x^{2}+21x-8x-21 $$Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -1 $ out of the last two terms.
$$ 8x^{2}+21x-8x-21 = x\left(8x+21\right) -1\left(8x+21\right) = \left(x-1\right) \left(8x+21\right) $$