\begin{align*} M''(0) & = n(n-1) ( p + (1-p))^{n-2} p^2 + n ( p + (1-p) )^{n-1} p \\ & = n(n-1)p^2 + np = (np)^2 + np - np^2 = np(1-p) + (np)^2. \end{align*}