<s>[INST] Prove $x^4+y^4+x^2+y^2+x^3y+y^3x\geq 0$ I tried 2 ways, first, take $x$ and $y$ $\geq0$ then obviously true. Take $x$ and $y$ both $\leq0$ same thing.
Now $y<0<x$ and wlog $|y|\leq x,$ then for big $x$ small $|y|$ we have $-x^4\leq x^3y$ and $-x^2\leq y^3x$ but if both $|y|$ and $x$ are big I got stuck.
So I decided to do it regular way and I looked for the maximum/maximas finding partial derivatives and setting it to zero and I got $$
\begin{cases}
  0=-4x^3+-2x+3x^2y-y^3 \\ 0=-4y^3-2y-3y^2x-x^3\\
\end{cases}
$$
easy to see (0,0) is a solution however I do not know how to prove that it is the only one
 [/INST] divide by $x^2y^2$if $x\neq0$ and $y\neq0$
$$\frac{x^2}{y^2}+\frac{y^2}{x^2}+\frac{1}{y^2}+\frac{1}{x^2}+\frac{x}{y}+\frac{y}{x}\geq 0$$
$$(\frac{x}{y}+0.5)^2+(\frac{y}{x}+0.5)^2+\frac{1}{y^2}+\frac{1}{x^2}-0.5\geq 0$$
now if $x\geq y$ then $(\frac{x}{y}+0.5)^2>1$
or if $y\geq x$ then $(\frac{y}{x}+0.5)^2>1$
</s>