Answer
$4p(p-2)(p+8)$
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 $GCF=
4p
,$ the given expression, $
4p^3+24p^2-64p
,$ is equivalent to
\begin{array}{l}\require{cancel}
4p(p^2+6p-16)
.\end{array}
In the expression above, the value of $c$ is $
-16
$ and the value of $b$ is $
6
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,-16\}, \{2,-8\}, \{4,-4\},
\{-1,16\}, \{-2,8\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-2,8
\}.$ Hence, the factored form of the given expression is
\begin{array}{l}\require{cancel}
4p(p-2)(p+8)
.\end{array}
