Answer
$-5a^3 \left( 1-2a+3a^{2} \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
First, get the $GCF$ of the given expression, $
-5a^3+10a^4-15a^5
.$ Then, divide the given expression and the $GCF.$ Express the answer as the product of the $GCF$ and the resulting quotient. Finally, factor out the $-1$.
$\bf{\text{Solution Details:}}$
The $GCF$ of the constants of the terms $\{
-5,10,-15
\}$ is $
5
.$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variable/s $\{
a^3,a^4,a^5
\}$ is $
a^3
.$ Hence, the entire expression has $GCF=
5a^3
.$
Factoring the $GCF=
5a^3
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
5a^3 \left( \dfrac{-5a^3}{5a^3}+\dfrac{10a^4}{5a^3}-\dfrac{15a^5}{5a^3} \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}
5a^3 \left( -a^{3-3}+2a^{4-3}-3a^{5-3} \right)
\\\\=
5a^3 \left( -a^{0}+2a^{1}-3a^{2} \right)
\\\\=
5a^3 \left( -1+2a-3a^{2} \right)
.\end{array}
Factoring out $-1$ from the second factor results to
\begin{array}{l}\require{cancel}
-5a^3 \left( 1-2a+3a^{2} \right)
.\end{array}
