I am trying to prove by mathematical induction $2\times1!+5\times2!+10\times3!+...+(n^2+1)n!=n(n+1)!$ for all positive integers $n$.

So far I have:

* Solved in the first case possible - $1$
* Assumed the function to be true for all positive integers
* Subbed in $k+1$ for $n$ leaving me with the following
$$((k+1)^2+1)(k+1)!=(k+1)(k+2)!$$

I am unsure where to go from here? Any help would be greatly appreciated!