Use properties of the scalar product to prove that for all vectors u,v $\displaystyle \in R^{2} $

$\displaystyle \lvert u+v \rvert^{2} + \lvert u-v \rvert^{2} = 2\lvert u \rvert^{2} + 2\lvert v \rvert^{2} $

i have expanded this got the above results, however I dont see how to use properties of the scalar product, i have simply applied some basic algebra?