Moment Generating Function of the Gamma Distribution

Let $X \sim \text{Gamma}(\alpha, \lambda)$ with PDF $f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}$ for $x > 0$. Derive the moment generating function $M_X(t) = E[e^{tX}]$ for $t < \lambda$. Then use $M_X(t)$ to compute $E[X]$ and $\text{Var}(X)$.