Answer
$(2-x)^2 \left( 1+2x \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Get the $GCF$ of the given expression, $
5(2-x)^2-2(2-x)^3
.$ 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 $\{
5,-2
\}$ is $
1
.$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variables $\{
(2-x)^2,(2-x)^3
\}$ is $
(2-x)^2
.$ Hence, the entire expression has $GCF=
(2-x)^2
.$
Factoring the $GCF=
(2-x)^2
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
(2-x)^2 \left( \dfrac{5(2-x)^2}{(2-x)^2}-\dfrac{2(2-x)^3}{(2-x)^2} \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}
(2-x)^2 \left( 5(2-x)^{2-2}-2(2-x)^{3-2} \right)
\\\\=
(2-x)^2 \left( 5(2-x)^{0}-2(2-x)^{1} \right)
\\\\=
(2-x)^2 \left( 5(1)-2(2-x) \right)
\\\\=
(2-x)^2 \left( 5-4+2x \right)
\\\\=
(2-x)^2 \left( 1+2x \right)
.\end{array}
