Answer
$4x(x-3)(5x+2)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
20x^3-52x^2-24x
,$ factor first the $GCF.$ Then find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Factoring the $GCF=
4x
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
4x(5x^2-13x-6)
.\end{array}
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
5(-6)=-30
$ and the value of $b$ is $
-13
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-15,2
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
4x(5x^2-15x+2x-6)
.\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}
4x[(5x^2-15x)+(2x-6)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
4x[5x(x-3)+2(x-3)]
.\end{array}
Factoring the $GCF=
(x-3)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
4x[(x-3)(5x+2)]
\\\\=
4x(x-3)(5x+2)
.\end{array}
