<s>[INST] $a+b+c=0;a^2+b^2+c^2=1$ then $a^4+b^4+c^4$ is equal to what? 
$a+b+c=0;a^2+b^2+c^2=1$ then $a^4+b^4+c^4$ is equal to what?

I tried to solve this problem, and I get $a^4+b^4+c^4 = 2(a^2b^2 + 1)$ but I'm not sure if it's  correct
 [/INST] $$a+b+c=0\Rightarrow c=-(a+b)$$
$$1=a^2+b^2+(a+b)^2=2a^2+2ab+2b^2=2(a^2+ab+b^2)\Rightarrow a^2+ab+b^2=\frac12$$
$$\begin{align*}
a^4+b^4+c^4 &= a^4+b^4+(a^4+4a^3b+6a^2b^2+4ab^3+b^4)\\
&= 2(a^4+2a^3b+3a^2b^2+2ab^3+b^4)\\
&= 2(a^2(a^2+ab+b^2)+b^2(a^2+ab+b^2)+ab(a^2+ab+b^2))\\
&= a^2+ab+b^2=\frac12
\end{align*}$$
</s>