\displaystyle \sum_{k=1}^{n}\frac{ k^2+k-1 }{(k+2)!}=\sum_{k=1}^{n}\frac{ k(k+2)-(k+1)}{ (k+2)!}=\sum_{k=1}^{n}\frac{ k}{(k+1)!}-\frac{(k+1)}{ (k+2)!}