Answer
$(a+2)(a+b)(a-b)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use factoring by grouping to factor the given expression, $
a^3+2a^2-ab^2-2b^2
.$ This would result to a factor that is a difference of $2$ squares. Use then the factoring of the difference of $2$ squares.
$\bf{\text{Solution Details:}}$
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(a^3+2a^2)-(ab^2+2b^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
a^2(a+2)-b^2(a+2)
.\end{array}
Factoring the $GCF=
(a+2)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(a+2)(a^2-b^2)
.\end{array}
The expressions $
a^2
$ and $
b^2
$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $
a^2-b^2
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(a+2)[(a)^2-(b^2)]
\\\\=
(a+2)[(a+b)(a-b)]
\\\\=
(a+2)(a+b)(a-b)
.\end{array}
