<s>[INST] Solve $x^2+xy+y^2+ \sqrt{3}y + 1=0$ for $xy$. I have a problem on my textbook:

$x,y \in\mathbb{R}$ and  $x^2+xy+y^2+ \sqrt{3}y + 1=0$ is given. Find the value of $xy$. 

I couldn't find.
 [/INST] Multiply both sides on 4
$$x^2+xy+y^2+ \sqrt{3}y + 1=0$$
$$4x^2+4xy+y^2+3y^2+ 4\sqrt{3}y + 4=0$$
$$(2x+y)^2+(\sqrt3y+2)^2=0$$
Then $y=-\frac2{\sqrt3}, x=\frac1{\sqrt3}$
$$xy=-\frac2{3}$$
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