Answer
$q^{10}-1$
Work Step by Step
Using $(a+b)(a-b)=a^2-b^2$ or the product of the sum and difference of like terms, the given expression, $
(q^5+1)(q^5-1)
$, is equivalent to
\begin{array}{l}\require{cancel}
(q^5)^2-(1)^2
\\\\=
q^{10}-1
.\end{array}