$\displaystyle \frac{(n+1)!}{n!} = \frac{n!(n+1)}{n!} = (n+1)$

I understand canceling, but I don't see how the initial fraction became $\displaystyle \frac{n!(n+1)}{n!}$.

Same with $\displaystyle \frac{(2n-1)!}{(2n+1)!} = \frac{(2n-1)!}{(2n-1)!(2n)(2n+1)} = \frac{1}{2n(2n+1)}$.