$\begin{aligned} & \text { Let } S=\sum_{n=1}^{-} \frac{2 n^2+n+1}{n !} \\ & =\bar{\sum}_{n=1}\left(\frac{2 n}{(n-1) !}+\frac{1}{(n-1) !}+\frac{1}{n !}\right) \\ & =\bar{\sum}_{n=1}\left(\frac{2}{(n-2) !}+\frac{3}{(n-1) !}+\frac{1}{n !}\right) \\ & =2\left(1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\ldots \infty\right) \\ & +3\left(1+\frac{1}{1 !}+\frac{1}{2 !}+\ldots \infty\right)+\left(\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\ldots\right) \\ & =2 e+3 e+e-1 \\ & =6 e-1 \\ & \end{aligned}$