Answer
$(z-5) \left( 2z+17 \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Get the $GCF$ of the given expression, $
(z-5)(z+7)+(z-5)(z+10)
.$ Divide the given expression and the $GCF.$ Express the answer as the product of the $GCF$ and the resulting quotient.
$\bf{\text{Solution Details:}}$
The $GCF$ of the constants of the terms $\{
1,1
\}$ is $
1
.$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variables $\{
(z-5)),(z-5)
\}$ is $
(z-5)
.$ Hence, the entire expression has $GCF=
(z-5)
.$
Factoring the $GCF=
(z-5)
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
(z-5) \left( \dfrac{(z-5)(z+7)}{(z-5)}+\dfrac{(z-5)(z+10)}{(z-5)} \right)
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
(z-5) \left( (z-5)^{1-1}(z+7)+(z-5)^{1-1}(z+10) \right)
\\\\=
(z-5) \left( (z-5)^{0}(z+7)+(z-5)^{0}(z+10) \right)
\\\\=
(z-5) \left( (1)(z+7)+(1)(z+10) \right)
\\\\=
(z-5) \left( z+7+z+10 \right)
\\\\=
(z-5) \left( 2z+17 \right)
.\end{array}