Answer
$2(8z-3)(6z-5)$
Work Step by Step
Let $a=4z-3.$ Then the given expression, $
6(4z-3)^2+7(4z-3)-3
,$ is equivalent to
\begin{array}{l}\require{cancel}
6a^2+7a-3
.\end{array}
The two numbers whose product is $ac=
6(-3)=-18
$ and whose sum is $b=
7
$ are $\{
9,-2
\}.$ Using these two numbers to decompose the middle term, then the factored form of $
6a^2+7a-3
$ is
\begin{array}{l}\require{cancel}
6a^2+9a-2a-3
\\\\=
(6a^2+9a)-(2a+3)
\\\\=
3a(2a+3)-(2a+3)
\\\\=
(2a+3)(3a-1)
.\end{array}
Since $a=4z-3$, then the expression $
(2a+3)(3a-1)
$ is equivalent to
\begin{array}{l}\require{cancel}
(2(4z-3)+3)(3(4z-3)-1)
\\\\=
(8z-6+3)(12z-9-1)
\\\\=
(8z-3)(12z-10)
\\\\=
(8z-3)2(6z-5)
\\\\=
2(8z-3)(6z-5)
.\end{array}
