By direct computation \begin{equation} \begin{split} a^2 + 2 a b + b^2 & = a a + 2 a b + b b \\ & = a a + a b + a b + b b \\ & = a a + a b + b a + b b \\ & = a (a + b) + b (a + b) \\ & = (a + b) (a + b) \\ & = (a + b)^2 \end{split} \end{equation} $\square$