\begin{gather*} \sigma^2 = E[X(X-1)] + \mu - \mu^2 \\ = \sum_{x=0}^{n} {x(x-1) \binom{n}{x} p^x (1-p)^{n-x}} + np - n^2p^2\\ = \sum_{x=2}^{n} {x(x-1) \frac{n(n-1)(n-2)!}{x(x-1)(x-2)!(n-x)!} p^x (1-p)^{n-x}} + np - n^2p^2\\ = n(n-1)p^2 \sum_{x=2}^{n} {\frac{(n-2)!}{(x-2)!((n-2)-(x-2))!} p^{x-2} (1-p)^{(n-2)-(x-2)}} + np - n^2p^2 \end{gather*}