I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$
becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$
and then how it becomes $$ \frac {n+n^2+n}{n-1} $$
I've tried applying the distributive property but I seem to be missing something.
Think of it like this:
$$\frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!}$$
Define $$D:=(n-1)!$$ then
$$\frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!}=\frac {nD+(n+1)nD}{nD-D}$$ now factor $D$
$$\frac {(n+(n+1)n)D}{(n-1)D}$$ now the D's cancel, $$\frac {(n+(n+1)n)}{(n-1)}$$ $$\frac {(n+n^2+n)}{(n-1)}$$ $$\frac {n+n^2+n}{n-1}$$ $$\frac {n^2+2n}{n-1}$$