\(x^3+y^3+3xy\left(x+y\right)-8-3xy\left(x+y\right)+6xy=0\)
\(\left(x+y\right)^3-8-3xy\left(x+y-2\right)=0\)
\(\left(x+y-2\right)\left(x^2+y^2+4+2xy+2x+2y\right)-3xy\left(x+y-2\right)=0\)
\(\left(x+y-2\right)\left(x^2+y^2-xy+2x+2y+4\right)=0\)
\(\left(x+y-2\right)\left[\left(x-\frac{y}{2}+1\right)^2+\frac{3}{4}\left(y^2-4y+4\right)\right]=0\)
\(\left(x+y-2\right)\left[\left(x-\frac{y}{2}+1\right)^2+\frac{3}{4}\left(y-2\right)^2\right]=0\)
\(\Rightarrow\left[{}\begin{matrix}x+y-2=0\\\left\{{}\begin{matrix}x-\frac{y}{2}+1=0\\y-2=0\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x+y=2\\\left\{{}\begin{matrix}x=0\\y=2\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow x+y=2\)