Answer
$(p+q+1)^2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
(p+q)^2+2(p+q)+1
,$ simplify the expression using substitution. The resulting expression is a perfect square trinomial. Use the factoring of perfect square trinomials. Finally, substitute back the original expression.
$\bf{\text{Solution Details:}}$
Let $z=
(p+q)
.$ The given expression becomes
\begin{array}{l}\require{cancel}
z^2+2z+1
.\end{array}
The trinomial above is a perfect square trinomial. Using $a^2\pm2ab+b^2=(a\pm b)^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(z+1)^2
.\end{array}
Since $z=
(p+q)
,$, then the expression above becomes
\begin{array}{l}\require{cancel}
((p+q)+1)^2
\\\\
(p+q+1)^2
.\end{array}