Answer
$2xy^3(x-12y)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
First, factor the GCF of the terms. Then, to factor the quadratic expression $x^2+bx+c,$ find two numbers, $m_1$ and $m_2,$ whose product is $c$ and whose sum is $b$. Express the factored form as $GCF(x+m_1)(x+m_2).$
$\bf{\text{Solution Details:}}$
Factoring the $GCF=
2xy^3
,$ the given expression, $
2x^3y^3-48x^2y^4+288xy^5
,$ is equivalent to
\begin{array}{l}\require{cancel}
2xy^3(x^2-24xy+144y^2)
.\end{array}
In the expression above, the value of $c$ is $
144
$ and the value of $b$ is $
-24
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,144\}, \{2,72\}, \{3,48\}, \{4,38\}, \{6,24\}, \{8,18\}, \{9,16\}, \{12,12\},
\{-1,-144\}, \{-2,-72\}, \{-3,-48\}, \{-4,-38\}, \{-6,-24\}, \{-8,-18\}, \{-9,-16\}, \{-12,-12\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-12,-12
\}.$ Hence, the factored form of the given expression is
\begin{array}{l}\require{cancel}
2xy^3(x-12y)(x-12y)
\\\\=
2xy^3(x-12y)^2
.\end{array}
