Answer
$a^7(5a+4)^2$
Work Step by Step
Factoring the $GCF=
a^7
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
40a^8+16a^7+25a^9
\\\\=
25a^9+40a^8+16a^7
\\\\=
a^7(25a^2+40a+16)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
a^7(25a^2+40a+16)
\end{array} has $ac=
25(16)=400
$ and $b=
40
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
20,20
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
a^7(25a^2+20a+20a+16)
.\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^7[(25a^2+20a)+(20a+16)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
a^7[5a(5a+4)+4(5a+4)]
.\end{array}
Factoring the $GCF=
(5a+4)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
a^7[(5a+4)(5a+4)]
\\\\=
a^7(5a+4)^2
.\end{array}
