Answer
$(p+8q-5)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
(p+8q)^2-10(p+8q)+25
,$ use substitution. Then use the factoring of trinomials in the form $x^2+bx+c.$
$\bf{\text{Solution Details:}}$
Let $z=(p+8q).$ Then the given expression is equivalent to
\begin{array}{l}\require{cancel}
z^2-10z+25
.\end{array}
In the trinomial expression above, $b=
-10
,\text{ and } c=
25
.$ Using the factoring of trinomials in the form $x^2+bx+c,$ the two numbers whose product is $c$ and whose sum is $b$ are $\left\{
-5,-5
\right\}.$ Using these two numbers, the factored form of the expression above is
\begin{array}{l}\require{cancel}
(z-5)(z-5)
\\\\=
(z-5)^2
.\end{array}
Since $z=(p+8q),$ by back substitution, the expression above is equivalent to
\begin{array}{l}\require{cancel}
(p+8q-5)^2
.\end{array}