# How do you factor (a-b)^2 - 16(a+2b)^2?

Sep 17, 2016

${\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2} = - 3 \left(a + 3 b\right) \left(5 a + 7 b\right)$

#### Explanation:

The difference of squares identity can be written:

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

Let $A = \left(a - b\right)$ and $B = 4 \left(a + 2 b\right)$

Then we find:

${\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2} = {\left(a - b\right)}^{2} - {\left(4 \left(a + 2 b\right)\right)}^{2}$

$\textcolor{w h i t e}{{\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2}} = \left(\left(a - b\right) - 4 \left(a + 2 b\right)\right) \left(\left(a - b\right) + 4 \left(a + 2 b\right)\right)$

$\textcolor{w h i t e}{{\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2}} = \left(a - b - 4 a - 8 b\right) \left(a - b + 4 a + 8 b\right)$

$\textcolor{w h i t e}{{\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2}} = \left(- 3 a - 9 b\right) \left(5 a + 7 b\right)$

$\textcolor{w h i t e}{{\left(a - b\right)}^{2} - 16 {\left(a + 2 b\right)}^{2}} = - 3 \left(a + 3 b\right) \left(5 a + 7 b\right)$