This is joint work with S.Galkin. Let X be a cubic hypersurface. We relate the geometry of the symmetric square Sym^2(X) to the Fano varietiy of lines F(X). We work in the Grothendieck ring of varieties, and then take motivic realizations to compare Hodge structure of F(X) to that of X. Time permitting, I will talk about relation to rationality of a cubic fourfold.