Answer
$(a^{3n}-1)(3a^{3n}+1)$
Work Step by Step
Using the factoring of trinomials in the form $ax^2+bx+c,$ the $\text{
expression
}$
\begin{array}{l}\require{cancel}
3a^{6n}-2a^{3n}-1
\end{array} has $ac=
3(-1)=-3
$ and $b=
-2
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-3,1
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
3a^{6n}-3a^{3n}+1a^{3n}-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}
(3a^{6n}-3a^{3n})+(a^{3n}-1)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
3a^{3n}(a^{3n}-1)+(a^{3n}-1)
.\end{array}
Factoring the $GCF=
(a^{3n}-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(a^{3n}-1)(3a^{3n}+1)
.\end{array}
