Question

In the following exercises, factor completely using the difference of squares pattern, if possible.

$a^2+6a+9-9b^2$

Short answer

The factored form of the given expression is$(a+3-3b)(a+3+3b)$.

Step-by-step solution

Step 1. Given information.

The given expression is:
$a^2+6a+9-9b^2$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}{a}^2+6a+9-9b^2& =& (a^2+2(a\left)\right(3)+3^2)-3^2{b}^2\\ & =& {(a+3)}^2-{\left(3b\right)}^2\left[\because {(a+b)}^2=a^2+2ab+b^2\right]\\ & =& (a+3-3b)(a+3+3b)\left[\because {a^2-b}^2=(a-b)(a+b)\right]\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(a+3-3b)(a+3+3b)$.