Answer
$z(z-4)(5z+1)$
Work Step by Step
Factoring the $GCF=
z
,$ the given $\text{
expression,
}
5z^3-19z^2-4z
,$ is equivalent to
\begin{align*}
z(5z^2-19z-4)
.\end{align*}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression
\begin{align*}
5z^2-19z-4
\end{align*} has $ac=
5(-4)=-20
$ and $b=
-19
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-20,1
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
z(5z^2-20z+z-4)
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
z[(5z^2-20z)+(z-4)]
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
z[5z(z-4)+(z-4)]
.\end{align*}
Factoring the $GCF=
(z-4)
$ of the entire expression above results to
\begin{align*}
&
z[(z-4)(5z+1)]
\\&=
z(z-4)(5z+1)
.\end{align*}
Hence, the factored form of $5z^3-19z^2-4z$ is $z(z-4)(5z+1)$.
