<s>[INST] Find all possible values of $ a^3 + b^3$ if $a^2+b^2=ab=4$. Find all possible values of $a^3 + b^3$ if $a^2+b^2=ab=4$.
From $a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(4-4)=(a+b)0$. Then we know $a^3+b^3=0$. 
If $a=b=0$, it is conflict with $a^2+b^2=ab=4$.
If $a\neq0$ and $b\neq0$, then $a$ and $b$ should be one positive and one negative. This contradict with $ab$=4.
I don't know the solution. 
Any help please.
 [/INST] Suppose there were real solutions to $a^2 + b^2 = ab = 4$. Then $$0 \leq (a - b)^2 = a^2 - 2ab + b^2 = (a^2 + b^2) - 2ab = -4$$ which is a a contradiction. Thus there cannot be real $a, b$ satisfying the requirements.
</s>