How to solve this summation without taylor?

Hint:  starting from OP's simplification, the following form will telescope nicely:

$$ \begin{align} \frac{n(n+1)^2}{ (n+2)!} &= \frac{(n+2)(n+1)n - (n+2)(n+1) + 2(n+2) -2}{(n+2)!} \\[5px] &= \frac{1}{(n-1)!} -\frac{1}{n!} + \frac{2}{(n+1)!} - \frac{2}{(n+2)!} \end{align} $$

As $n(n+1)^2$ is $O(n^3),$

write $n(n+1)^2=(n+2)(n+1)n+ A(n+2)(n+1)+B(n+2)+C$

so that $$\dfrac{n(n+1)^2}{(n+2)!}=\dfrac1{(n-1)!}+\dfrac A{n!}+\dfrac B{(n+1)!}+\dfrac C{(n+2)!}$$

$n=-2\implies C=(-2)(-2+1)^2=-2$

Comparing the coefficients of $n^2,$ $$2=3+A\iff A=-1$$

Comparing the coefficients of $n,$ $$1=2+3A+B\iff B=1-2-3A=2$$

Interestingly, this leads to a Telescoping series,

otherwise, we had to use $$e^y=\sum_{r=0}^\infty\dfrac{y^r}{r!}$$