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E[x^2]=\sum^\infty_{k=1}k^2e^{-\lambda}*\frac{\lambda^k}{k!}\\ =\lambda e^{-\lambda}\sum^{\infty}_{k=1}k*\frac{\lambda^{k-1}}{(k-1)!} =\lambda e^{-\lambda}\sum^{\infty}_{k=0}(k+1)\frac{\lambda^k}{k!}\begin{Bmatrix}\text{replace}\\k\larr k+1\end{Bmatrix}\\ =\lambda e^{-\lambda}(\sum^\infty_{k=0}k\frac{\lambda^k}{k!}+e^{\lambda}\sum^\infty_{k=0}\frac{\lambda^k}{k!})\\ =\lambda(\lambda+e^{-\lambda}e^{\lambda})\\ =\lambda(\lambda + 1)\\ =\lambda^2+\lambda