Answer
$(2xz-1)(3xz+4)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
In the given expression, $
6x^2z^2+5xz-4
,$ the value of $ac$ is $
6(-4)=-24
$ and the value of $b$ is $
5
.$
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 $\{
-3,8
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
6x^2z^2-3xz+8xz-4
.\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}
(6x^2z^2-3xz)+(8xz-4)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3xz(2xz-1)+4(2xz-1)
.\end{array}
Factoring the $GCF=
(2xz-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2xz-1)(3xz+4)
.\end{array}
