$F=5x^2+4xy+y^2+8x+100$
$F=(4x^2+4xy+y^2)+(x^2+8x+16)+84$
$F=(2x+y)^2+(x+4)^2+84$
Vì $(2x+y)^2+(x+4)^2 \ge 0\; \forall x\in \mathbb{R}$
$\Rightarrow (2x+y)^2+(x+4)^2+84 \ge 84\; \forall x\in \mathbb{R}$
Vậy $\min F=84$ khi $\begin{cases}2x+y=0\\x+4=0\end{cases}\Leftrightarrow \begin{cases}x=-4\\y=8\end{cases}$
$E=2x^2-2xy+y^2+12x-4y$
$E=(x^2-2xy+y^2)+(4x-4y)+4+(x^2+8x+16)-20$
$E=[(x-y)^2+4(x-y)+4]+(x+4)^2-20$
$E=(x-y+2)^2+(x+4)^2-20$
Vì $(x-y+2)^2+(x+4)^2 \ge 0\; \forall x\in \mathbb{R}$
$\Rightarrow (x-y+2)^2+(x+4)^2-20 \ge -20\; \forall x\in \mathbb{R}$
Vậy $\min F=-20$ khi $\begin{cases}x-y+2=0\\x+4=0\end{cases}\Leftrightarrow \begin{cases}x=-4\\y=-2\end{cases}$
