Is this (or when) does this equality hold?

Question

Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers

$$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$

Is this always an equality? Under what circumstances does this equality hold?

Answer 1

The squares are obviously equal. What might cause a problem is the last term.

So, the equality holds iff $(ax+by)(cx+dy)\geq0$