# How do you factor (b - 3)^2 - (a + 1)^2 ?

Jun 22, 2015

This is a difference of squares, so it factors as

${\left(b - 3\right)}^{2} - {\left(a + 1\right)}^{2}$

$= \left(\left(b - 3\right) - \left(a + 1\right)\right) \left(\left(b - 3\right) + \left(a + 1\right)\right)$

$= \left(b - a - 4\right) \left(b + a - 2\right)$

#### Explanation:

For any $p$ and $q$, we have:

${p}^{2} - {q}^{2} = \left(p - q\right) \left(p + q\right)$

Putting $p = \left(b - 3\right)$ and $q = \left(a + 1\right)$, we have:

${\left(b - 3\right)}^{2} - {\left(a + 1\right)}^{2}$

$= {p}^{2} - {q}^{2}$

$= \left(p - q\right) \left(p + q\right)$

$= \left(\left(b - 3\right) - \left(a + 1\right)\right) \left(\left(b - 3\right) + \left(a + 1\right)\right)$

$= \left(b - a - 4\right) \left(b + a - 2\right)$