Answer
$a^6(3a+4)(3a+2)$
Work Step by Step
Factoring the $GCF=
a^6
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
18a^7+8a^6+9a^8
\\\\=
9a^8+18a^7+8a^6
\\\\=
a^6(9a^2+18a+8)
.\end{array}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
a^6(9a^2+18a+8)
\end{array} has $ac=
9(8)=72
$ and $b=
18
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
12,6
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
a^6(9a^2+12a+6a+8)
.\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^6[(9a^2+12a)+(6a+8)]
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
a^6[3a(3a+4)+2(3a+4)]
.\end{array}
Factoring the $GCF=
(3a+4)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
a^6[(3a+4)(3a+2)]
\\\\=
a^6(3a+4)(3a+2)
.\end{array}