# How do you factor this?

## $12 {\left(x + 2\right)}^{2} - 14 \left(x + 2\right) - 6$

Dec 15, 2017

$= 2 \left(2 x + 1\right) \left(3 x + 7\right)$

#### Explanation:

let $\left(x + 2\right) = a$ for now.

$= 12 {a}^{2} - 14 a - 6$ Factor out 2
$= 2 \left(6 {a}^{2} - 7 a - 3\right)$ Factor by grouping - Break middle term
$= 2 \left(6 {a}^{2} + 2 a - 9 a - 3\right)$ Factor
$= 2 \left(2 a \left(3 a + 1\right) - 3 \left(3 a + 1\right)\right)$ Factor out $\left(3 a + 1\right)$
$= 2 \left(2 a - 3\right) \left(3 a + 1\right)$

substitute $\left(x + 2\right)$ for $a$

$= 2 \left(2 \left(x + 2\right) - 3\right) \left(3 \left(x + 2\right) + 1\right)$ Simplify by distribution
$= 2 \left(2 x + 4 - 3\right) \left(3 x + 6 + 1\right)$ Add/Subtract
$= 2 \left(2 x + 1\right) \left(3 x + 7\right)$
Done!

Lets test a value for fun.
for $x = 1$,
$= 12 {\left(1 + 2\right)}^{2} - 14 \left(1 + 2\right) - 6$
$= 12 \left(9\right) - 14 \left(3\right) - 6$
$= 108 - 42 - 6$
$= 60$

Now for the factored version.

=2(2(1)+1)(3(1)+7))
$= 2 \left(3\right) \left(10\right)$
$= 60$

$60 = 60$, $L S = R S$