Answer
$(9z-1)(3z+5)$
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, $
27z^2+42z-5
,$ the value of $ac$ is $
27(-5)=-135
$ and the value of $b$ is $
42
.$ The $2$ numbers that have a product $ac$ and a sum of $b$ are $\{
-3,45
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
27z^2-3z+45z-5
.\end{array}
Grouping the first and third terms and the second and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(27z^2-3z)+(45z-5)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3z(9z-1)+5(9z-1)
.\end{array}
Factoring the $GCF=
(9z-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(9z-1)(3z+5)
.\end{array}
