Answer
$a^2b(ab+5)(ab-3)$
Work Step by Step
Factoring the $GCF=
a^2b
$, then the given expression, $
a^4b^3+2a^3b^2-15a^2b
$ is equivalent to
\begin{array}{l}
a^2b(a^2b^2+2ab-15)
.\end{array}
The two numbers whose product is $ac=
1(-15)=-15
$ and whose sum is $b=
2
$ are $\{
5,-3
\}$. Using these two numbers to decompose the middle term of the expression, $
a^2b(a^2b^2+2ab-15)
,$ then the factored form is
\begin{array}{l}
a^2b(a^2b^2+5ab-3ab-15)
\\\\=
a^2b[(a^2b^2+5ab)-(3ab+15)]
\\\\=
a^2b[ab(ab+5)-3(ab+5)]
\\\\=
a^2b[(ab+5)(ab-3)]
\\\\=
a^2b(ab+5)(ab-3)
.\end{array}
