Making use of the method of differences, prove that $\displaystyle \sum_{n = 1}^{k}\frac{n^2+6n+4}{(n+2)!}=3-\frac{1}{(k+1)!}-\frac{4}{(k+2)!} $

$\displaystyle \sum_{n = 1}^{k}\frac{n^2+6n+4}{(n+2)!}=\sum_{n=1}^{k}\frac{ (n+2)(n+1)+3(n+2)-4}{(n+2)!}$

$\displaystyle =\sum_{n = 1}^{k}\frac{(n+2)(n+1)}{(n+2)!}+\frac{3(n+2)}{(n+2 )!}-\frac{4}{(n+2)!}$

$\displaystyle =\sum_{n = 1}^{k}\frac{(n+2)(n+1)}{(n+2)(n+1)n!}+\frac{3(n+2) }{(n+2)(n+1)!}-\frac{4}{(n+2)!}$

$\displaystyle =\sum_{n = 1}^{k}\frac{1}{n!}+\frac{3}{(n+1)!}-\frac{4}{n+2)!}$