Inequality in Banach space

Question

So I have to either prove or disprove this inequality:

$$ \left\lVert x\right\rVert^2 - \left\lVert y\right\rVert^2 \le \left\lVert x-y\right\rVert \left\lVert x+y\right\rVert$$

I know this to be true in Hilbert space. But i cant figure out what to do in Banach space.