Answer
$a(5a-1)^2$
Work Step by Step
Factoring the $GCF=
a
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
a-10a^2+25a^3
\\\\=
a(1-10a+25a^2)
\\\\=
a(25a^2-10a+1)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
a(25a^2-10a+1)
\end{array} has $ac=
25(1)=25
$ and $b=
-10
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-5,-5
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
a(25a^2-5a-5a+1)
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
a[(25a^2-5a)-(5a-1)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
a[5a(5a-1)-(5a-1)]
.\end{array}
Factoring the $GCF=
(a^{n}-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
a[(5a-1)(5a-1)]
\\\\=
a(5a-1)^2
.\end{array}
