Solve left(x-yright)left(x^2+y^2+4xyright)=2

$$\left( x-y \right) \left( { x }^{ 2 } + { y }^{ 2 } +4xy \right) =2$$

Solve for x

$$x=-\sqrt{3}i\left(\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1\right)^{-\frac{1}{3}}y^{2}+\frac{\sqrt{3}i\sqrt[3]{\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1}}{2}-\frac{\sqrt[3]{\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1}}{2}-y-\left(\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1\right)^{-\frac{1}{3}}y^{2}$$

$$x=2\left(\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1\right)^{-\frac{1}{3}}y^{2}+\sqrt[3]{\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1}-y$$

$$x=\sqrt{3}i\left(\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1\right)^{-\frac{1}{3}}y^{2}-\frac{\sqrt{3}i\sqrt[3]{\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1}}{2}-\frac{\sqrt[3]{\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1}}{2}-y-\left(\sqrt{1-4y^{3}-4y^{6}}-2y^{3}+1\right)^{-\frac{1}{3}}y^{2}$$

Solve for y

$$\left\{\begin{matrix}y=-\sqrt{3}i\left(\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1\right)^{-\frac{1}{3}}x^{2}+\frac{\sqrt{3}i\sqrt[3]{\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1}}{2}-\frac{\sqrt[3]{\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1}}{2}-x-\left(\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1\right)^{-\frac{1}{3}}x^{2}\text{; }y=2\left(\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1\right)^{-\frac{1}{3}}x^{2}+\sqrt[3]{\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1}-x\text{; }y=\sqrt{3}i\left(\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1\right)^{-\frac{1}{3}}x^{2}-\frac{\sqrt{3}i\sqrt[3]{\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1}}{2}-\frac{\sqrt[3]{\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1}}{2}-x-\left(\sqrt{1+4x^{3}-4x^{6}}-2x^{3}-1\right)^{-\frac{1}{3}}x^{2}\text{, }&x\neq 0\\y=-\sqrt[3]{2}\text{; }y=\sqrt[3]{-2}\text{; }y=\sqrt[3]{2}e^{\frac{5\pi i}{3}}\text{, }&x=0\end{matrix}\right.$$