Sharp(er) bound on a ratio between factorials

Question

Let $p$, $N$ be integers and let $N > 0$. I have the ratio $$ R(N,p)=\frac{(N+p)!}{(N-1)!}=(N+p)(N+p-1)\cdots (N+1)N. $$ It is rather easy to prove by induction that $$ R(N,p) \leq (p+1)!N^{p+1}. $$ In order to prove this, though, one bounds $N^k \leq N^{k+1}$ which does not appear to be tight. For example $$ R(N,1)=(N+1)N = N^2+N\leq 2N^2. $$ I am looking for a tighter bound, possibly of the form $$ R(N,p)\leq C(p)N^{p+1}. $$ Is it possible to obtain one?