How do I prove that a term is divisible by another term by using induction?

Question

I need to prove that $\frac{(2n)!}{n!n!}$ is divisble by $(n+1)$, for all natural numbers $n$.
I ofcourse start with the base case $n = 1$ and formulate the induction hypothesis that $\frac{(2n)!}{n!n!} = m(n + 1)$ for some integer $m$.
Then I perform the induction step: $\frac{(2n + 2)!}{(n + 1)!(n + 1)!} = l(n + 2)$
Then I expand the expression to $\frac{(2n + 2)(2n + 1)(2n)!}{(n + 1)(n + 1)(n!)(n!)} = l(n + 2)$ and then substitute
$\frac{(2n + 2)(2n + 1)(n + 1)m}{(n + 1)(n + 1)} = l(n + 2)$. I further simplify and arrive at $2m(2n + 1) = l(n + 2)$ and this is where I am stuck. I don't know what to do next or if I started wrong. Any help would be appreciated. (And excuse me if I miss used some terms, I am not a native English speaker.)