Prove the identity $1(1!)+2(2!)+3(3!)+...+n(n!)=(n+1)!-1$ without using induction

Question

I was asked to prove this identity using two methods. I have already completed a proof by induction, but I don't know another method to prove this and the hint I was provided doesn't make any sense to me.

The hint provided was: For the other method, try to express $k(k!)$ as the difference of factorials.