How do I go about proving the following statement?
Let $a, b \in \mathbb{R}$. Let the inequality $$|(ax + by)(ay + bx)| \le x^2 + y^2$$ hold $\forall x, y \in \mathbb{R}$. Show that $$a^2 + b^2 \le 2$$
I have been going at this problem for an hour now and I can't seem to find a reasonable approach, I always have a "leftover" term that I can't get rid of.