[tex]x^{3}+y^{3}+3(x^{2}+y^{2})+4(x+y)+4=0\Leftrightarrow (x+1)^{3}+(y+1)^{3}+(x+y+2)=0\Leftrightarrow (x+y+2)\left [ (x+1)^{2}-(x+1)(y+1)+(y+1)^{2} \right ]+(x+y+2)=0\Leftrightarrow (x+y+2)\left [ (x+1)^{2}-(x+1)(y+1)+(y+1)^{2}+1 \right ]=0\Leftrightarrow x+y+2=0\Leftrightarrow x+y=-2[/tex]
[tex]A=\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}\leq \frac{x+y}{\frac{(x+y)^{2}}{4}}=-2[/tex]
dấu = xảy ra <=>x=y=-1