Answer
$(x^2+2x+3)(a+1)$
Work Step by Step
Using factoring by grouping, the factored form of the given expression, $
ax^2+2ax+3a+x^2+2x+3
,$ is
\begin{array}{l}\require{cancel}
(ax^2+2ax+3a)+(x^2+2x+3)
\\\\=
a(x^2+2x+3)+(x^2+2x+3)
\\\\=
(x^2+2x+3)(a+1)
.\end{array}