\mathrm{E}\left(e^{2X}\right)=\sum\limits_{n=0}^{\infty}{e^{2n}\cdot\mathbb{P}\left(X=n\right)}=\sum\limits_{n=0}^{\infty}{e^{2n}\cdot\frac{\lambda^n}{n!}e^{-\lambda}}=\ \left[\lambda=1\right]\ =