Answer
$4(a-b^2)(a^2+2b)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Factor the $GCF$ of the terms. Then, group the terms of the given expression, $
4a^3-4a^2b^2+8ab-8b^3
,$ such that the factored form of the groupings will result to a factor common to the entire expression. Then, factor the $GCF$ in each group. Finally, factor the $GCF$ of the entire expression.
$\bf{\text{Solution Details:}}$
Using the $GCF=
4
,$, the expression above is equivalent to
\begin{array}{l}\require{cancel}
4\left(\dfrac{4a^3}{4}-\dfrac{4a^2b^2}{4}+\dfrac{8ab}{4}-\dfrac{8b^3}{4}\right)
\\\\=
4\left(a^3-a^2b^2+2ab-2b^3\right)
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
4[(a^3-a^2b^2)+(2ab-2b^3)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
4[a^2(a-b^2)+2b(a-b^2)]
.\end{array}
Factoring the $GCF=
(a-b^2)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
4[(a-b^2)(a^2+2b)]
\\\\=
4(a-b^2)(a^2+2b)
.\end{array}
