<s>[INST] Find parameter of inequality Find the parameter $m \in \mathbb R$ so that  $$x^2 - 2xy + 2y^2 + 2x - 6y + m \ge 0, \forall x, y \in \mathbb R$$
I tried forming some squares but it seems it doesn't help me
$$(x-y)^2 + y^2 + 2x -6y + m \ge 0$$
 [/INST] $y^2+x^2+1+2x-2y-2xy=(y-x-1)^2$ and $y^2-4y+4=(y-2)^2$. Total of these, we yields
$$x^2 - 2xy + 2y^2 + 2x - 6y + m = (y-x-1)^2 + (y-2)^2 + m - 5 \geq 0$$
Hence, we find that $ m\geq 5$.  
</s>