Answer
$a^2+4a+4$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the product of $2$ binomials, the given expression, $
(a+2)(a+2)
$, is equivalent to
\begin{array}{l}\require{cancel}
a(a)+a(2)+2(a)+2(2)
\\\\=
a^2+2a+2a+4
\\\\=
a^2+4a+4
.\end{array}