Help me figure out how to come up with the start of this proof on my own?

Question

So the statement is $$4 + \sin^2x \geq 6\sin2x,\quad\forall x \in \Bbb R.$$ Now the start of this proof says

"We know that $(3\sin x - 2\cos^2x)^2 \geq 0$."

And then they just expanded this to get the starting statement. I get why this works but how the heck do you come up with the beginning statement?

Thanks!