How to know the positivity of a given polynomial

Question

Let $f(x)=\dfrac{1}{(2p+1)(2p+2)}x^{2p}+\sum\limits_{i=1}^{2p}\binom{2p+1}{i}\dfrac{1}{i+1}x^{i-1}.$ How do you know about the positivity of the $f(x)$? I am trying to induction on $p$ as well as combine the term $1,x, x^2$ and $1,x^3, x^6$, etc as well as combine the terms $1,x,x^2$ and $x^2, x^3, x^4$, etc to get the sum of squares. However, it is hard to control when the coefficient depends on the binomial with $p$. I also tried to use Sturm theorem as well as Eneström's theorem to determine its zeros points and from there to see the positivity but I am not successful either. It would be great if someone can give me some help. Thanks.