## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

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