Define $g(\epsilon) = \langle x+\epsilon y , x + \epsilon y\rangle$ where $\epsilon$ is real. Then $g(\epsilon) = \langle x,x\rangle + \epsilon (\langle y,x\rangle + \langle x,y\rangle) + \epsilon^2 \langle y,y\rangle .$ Hence, $$ \Vert x + y \Vert ^2 + \Vert x - y \Vert ^2 = g(1) + g(-1) = 2\langle x,x\rangle + 2\langle y,y\rangle = 2 \Vert x \Vert ^2 + 2 \Vert y \Vert ^2.$$