Logic Tip: The AM-GM inequality provides a blazing fast shortcut! For positive numbers $a$ and $b$, $\frac{a+b}{2} \ge \sqrt{ab}$. Let $a=x$ and $b=2y$. Then $\frac{x+2y}{2} \ge \sqrt{2xy}$. Since $x+2y=8$, we have $4 \ge \sqrt{2xy}$. Squaring gives $16 \ge 2xy$, meaning $xy \le 8$. The maximum is clearly 8.