Bounding $H(x)=x\log \frac{x}{p} + (1-x)\log\frac{1-x}{1-p} $

Question

I am trying to demonstrate that $H(p+x)\geq 2x^2$, where $H(x)=x\log \frac{x}{p} + (1-x)\log\frac{1-x}{1-p} $ and $p,x\in (0,1)$, $p<p+x<1$. Any hint? Thank you very much!