Answer
$3m^2(m-4)(m+6)$
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 $GCF=
3m^2
,$ the given expression, $
3m^4+6m^3-72m^2
,$ is equivalent to
\begin{array}{l}\require{cancel}
3m^2(m^2+2m-24)
.\end{array}
In the expression above, the value of $c$ is $
-24
$ and the value of $b$ is $
2
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,-24\}, \{2,-12\}, \{3,-8\}, \{4,-6\},
\{-1,24\}, \{-2,12\}, \{-3,8\}, \{-4,6\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-4,6
\}.$ Hence, the factored form of the given expression is
\begin{array}{l}\require{cancel}
3m^2(m-4)(m+6)
.\end{array}