Binomial Moments via the Moment Generating Function

Let $X \sim \text{Binomial}(n, p)$.

(a) Derive the moment generating function $M_X(t) = E[e^{tX}]$ in closed form.

(b) By differentiating $M_X(t)$ and evaluating at $t = 0$, find $E[X]$ and $E[X^2]$, and hence $\text{Var}(X)$.