# How do you factor  a^2 + ab -4 -4b by grouping?

May 26, 2017

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

#### Explanation:

Strategy: Factoring by grouping usually involves four terms, which we are given.

Step 1. Factor out anything common to the first two terms.
$\textcolor{b l u e}{{a}^{2} + a b} - 4 - 4 b$
$a \left(a + b\right) - 4 - 4 b$

Step 2. Factor out anything common to the second two terms.
$a \left(a + b\right) \textcolor{red}{- 4 - 4 b}$
$a \left(a + b\right) - 4 \left(1 + b\right)$

Note this doesn't help us very much because we can't do a third step of factoring out the expressions in parenthesis. So, the next strategy is to rearrange the original terms and see if that works better.

Step 1'. Rearrange the terms so that $b$'s are next to each other.
${a}^{2} - 4 + a b - 4 b$

Step 2'. Factor out the $b$ terms on the right
${a}^{2} - 4 + a \textcolor{b l u e}{b} - 4 \textcolor{b l u e}{b}$
${a}^{2} - 4 + \textcolor{b l u e}{b} \left(a - 4\right)$

Step 3'. Factor out the terms on the left
$\left(a + 2\right) \left(a - 2\right) + b \left(a - 4\right)$

Although this arguably looks neater than the previous solution, this expression cannot be factored any further. This is the final answer.

Note that if the question had been to factor by grouping the expression ${a}^{2} + a b - 4 - \textcolor{red}{2} b$, then we would have been able to factor more completely to
${a}^{2} - 4 + a b - 2 b$
$\left(a + 2\right) \left(a - 2\right) + b \left(a - 2\right)$
$\left(a + 2 + b\right) \left(a - 2\right)$