Trouble solving $[\frac{np+1}{(n+2)}-p]^2+\frac{np(1-p)}{(n+2)^2}<\frac{(1-p) p}{n}$

Question

After extending the fraction I've got the following form

$$\frac{n - 4p +4p^2+8np^2-8np}{n(n + 2)^2}<0$$

but I'm not able to separate the variables and find what should be the solution

$$n>0,\ \frac{1}{2} \left(1 - \sqrt{\frac{n+1}{2n+1}}\right)<p<\frac{1}{2} \left(\sqrt{\frac{n+1}{2n+1}} + 1\right)$$