<s>[INST] Given that $a+b+c=0$, show that $2(a^4+b^4+c^4)$ is a perfect square Given that $a+b+c=0$. Show that: $2(a^4+b^4+c^4)$ is a perfect square
MY ATTEMPTS:
I found that when $a+b+c=0$, $a^3+b^3+c^3=3abc$
So I did:
$(a^3+b^3+c^3)(a+b+c)$ -- $a^4+b^4+c^4=-(a^3c+a^3b+ab^3+b^3c+c^3a+c^3b)$
And then I tried to substitute $a^4+b^4+c^4$, but I found nothing that I thought relevant to the question
 [/INST] Square $a+b+c=0$
\begin{eqnarray*}
a^2+b^2+c^2=-2(ab+bc+ca).
\end{eqnarray*}
Square this
\begin{eqnarray*}
a^4+b^4+c^4+2(a^2b^2+b^2c^2+c^2a^2)=4(a^2b^2+b^2c^2+c^2a^2)+8abc(a+b+c)
\end{eqnarray*}
The last term is zero ... rearrange 
\begin{eqnarray*}
2(a^4+b^4+c^4)=4(a^2b^2+b^2c^2+c^2a^2)=(a^2+b^2+c^2)^2.
\end{eqnarray*}
the last equality follows from the first equation.
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