Show that $\u27e8x+y,x-y\u27e9=\Vert x{\Vert}_{2}^{2}-\Vert y{\Vert}_{2}^{2}$?

given $x,y\in {\mathbb{R}}^{n}$ I am trying to prove this:

$\u27e8x-y,x-y\u27e9=\u27e8x,x\u27e9-2\u27e8x,y\u27e9+\u27e8y,y\u27e9=\Vert x{\Vert}_{2}^{2}-2\u27e8x,y\u27e9+\Vert y{\Vert}_{2}^{2}.$

What I tried is using the fact that:

$\u27e8x-y,x-y\u27e9=\u27e8x,x\u27e9-2\u27e8x,y\u27e9+\u27e8y,y\u27e9=\Vert x{\Vert}_{2}^{2}-2\u27e8x,y\u27e9+\Vert y{\Vert}_{2}^{2}.$

Then, since y=−(−y), I have

$\u27e8x+y,x-y\u27e9=\u27e8x-(-)y,x-y\u27e9$

But I don't know how I proceed from this... Maybe my approach is wrong?

given $x,y\in {\mathbb{R}}^{n}$ I am trying to prove this:

$\u27e8x-y,x-y\u27e9=\u27e8x,x\u27e9-2\u27e8x,y\u27e9+\u27e8y,y\u27e9=\Vert x{\Vert}_{2}^{2}-2\u27e8x,y\u27e9+\Vert y{\Vert}_{2}^{2}.$

What I tried is using the fact that:

$\u27e8x-y,x-y\u27e9=\u27e8x,x\u27e9-2\u27e8x,y\u27e9+\u27e8y,y\u27e9=\Vert x{\Vert}_{2}^{2}-2\u27e8x,y\u27e9+\Vert y{\Vert}_{2}^{2}.$

Then, since y=−(−y), I have

$\u27e8x+y,x-y\u27e9=\u27e8x-(-)y,x-y\u27e9$

But I don't know how I proceed from this... Maybe my approach is wrong?