<s>[INST] $x^2 + y^2+xy = 1$ , then find the minimum of $x^3 y + xy^3 +4$ x and y belongs to real numbers. $ x^2 + y^2+xy = 1 $.  then find the minimum value of $x^3 y + xy^3 +4$.
I assume  $ x = r \sin (w)$ and $ y = r\cos(w) $. $  x^3 y + xy^3 +4 = L $  which give me $ \frac{2}{3} \le r^2 \le 2 $ I am stuck after that.Its my  Humble request to help me after that.
 [/INST] Hint: $x^2+y^2=1-xy$, so that $x^3y+xy^3+4=xy(x^2+y^2)+4=xy(1-xy)+4$. Substitute $z=xy$. What do you end up with?
</s>