I was reading through a proof for $-1 \leq \rho \leq 1$, where $\rho$ is correlation coefficient, and I became a little confused by the motivation in the first step.
What was the motivation to start the proof with $E((V+tW)^2)$ where $V = X - \mu_X$ and $W = Y - \mu_Y$? That is, why did they multiply $W$ by $t$?
"It works out in the end" is a pretty good motivation if you ask me.
As to how the technique could be imagined for the first time, we want to know whether $$ |E(XY)|\leq \sqrt{E(X^2)E(Y^2)}\\ E(XY)^2-E(X^2)E(Y^2)\leq0 $$ This is the discriminant of some quadratic polynomial (it's easier to see if you multiply both sides by $4$) and translating that inequality into a property of said polynomial is easy. So, which polynomial could that be, and what can we do with it?