It's used to perform a sort of trick. Adding the same number to both sides of an inequality preserves the inequality, so there is nothing wrong with adding the 4ab to both sides. The nice this is, it converts (a-b)^2 on the left side into (a+b)^2 and that turns out to be very useful later on in the proof. If your asking how somebody could come up with the proof listed there, I can't answer that. Often, once someone has come up with a proof of something, they will try to find a better proof. Once you know something is true, you can spend all of your time being clever. It's sort of like machismo.