Prove that {IIxII + IIyII}^2 = IIxII^2 + 2*IIxII*IIyII + IIyII^2 if x and y are elements of a vector space R!?
Well that's very easy.
Let the norm of $\displaystyle x$ be definied such as $\displaystyle \|x\| := \sqrt{\langle x,x\rangle}$ where $\displaystyle \langle , \rangle$ is an inner product.
So you have $\displaystyle \left (\sqrt{\langle x,x\rangle}+\sqrt{\langle y,y\rangle}\right)^2$. It is simply equal to $\displaystyle \langle x,x\rangle+2\cdot \sqrt{\langle x,x\rangle}\cdot \sqrt {\langle y,y\rangle}+ \langle y,y\rangle$ which is equal to what you had to prove.