# How do you factor -6cf + 8c^2d - 9df + 12cd^2?

Jan 1, 2017

#### Answer:

$- 6 c f + 8 {c}^{2} d - 9 \mathrm{df} + 12 c {d}^{2} = \left(2 c + 3 d\right) \left(4 c d - 3 f\right)$

#### Explanation:

Given:

$- 6 c f + 8 {c}^{2} d - 9 \mathrm{df} + 12 c {d}^{2}$

Note that the ratio of the first and second terms is the same as that between the third and fourth terms. So this cubic will factor by grouping:

$- 6 c f + 8 {c}^{2} d - 9 \mathrm{df} + 12 c {d}^{2} = \left(- 6 c f + 8 {c}^{2} d\right) + \left(- 9 \mathrm{df} + 12 c {d}^{2}\right)$

$\textcolor{w h i t e}{- 6 c f + 8 {c}^{2} d - 9 \mathrm{df} + 12 c {d}^{2}} = 2 c \left(- 3 f + 4 c d\right) + 3 d \left(- 3 f + 4 c d\right)$

$\textcolor{w h i t e}{- 6 c f + 8 {c}^{2} d - 9 \mathrm{df} + 12 c {d}^{2}} = \left(2 c + 3 d\right) \left(4 c d - 3 f\right)$

The remaining quadratic term is irreducible.