# How do you factor 4x^3 + 2x^2 - 24x - 12?

May 22, 2016

$4 {x}^{3} + 2 {x}^{2} - 24 x - 12$

$= 2 \left({x}^{2} - 6\right) \left(2 x + 1\right)$

$= 2 \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right) \left(2 x + 1\right)$

#### Explanation:

Notice that the ratio of the coefficients of the first and second terms is the same as the ratio of the coefficients of the third and fourth terms, so this cubic can be factored by grouping:

$4 {x}^{3} + 2 {x}^{2} - 24 x - 12$

$= \left(4 {x}^{3} + 2 {x}^{2}\right) - \left(24 x + 12\right)$

$= 2 {x}^{2} \left(2 x + 1\right) - 12 \left(2 x + 1\right)$

$= \left(2 {x}^{2} - 12\right) \left(2 x + 1\right)$

$= 2 \left({x}^{2} - 6\right) \left(2 x + 1\right)$

$= 2 \left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right) \left(2 x + 1\right)$