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 $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -6} $.
$$ a \cdot c = -12 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -12 $ and add to $ b = -1 $.
Step 4: All pairs of numbers with a product of $ -12 $ are:
PRODUCT = -12 | |
-1 12 | 1 -12 |
-2 6 | 2 -6 |
-3 4 | 3 -4 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = -1 }$
PRODUCT = -12 and SUM = -1 | |
-1 12 | 1 -12 |
-2 6 | 2 -6 |
-3 4 | 3 -4 |
Step 6: Replace middle term $ -1 x $ with $ 3x-4x $:
$$ 2x^{2}-x-6 = 2x^{2}+3x-4x-6 $$Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -2 $ out of the last two terms.
$$ 2x^{2}+3x-4x-6 = x\left(2x+3\right) -2\left(2x+3\right) = \left(x-2\right) \left(2x+3\right) $$