<s>[INST] Understanding $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ by means of determinants. The identity comes from expanding the determinant
$$\begin{vmatrix}
a & b & c \\
c & a & b \\
b & c & a \\
\end{vmatrix}$$
in two ways.
The LHS comes from expanding the determinant by Sarrus' rule. The RHS comes from adding up all columns to the first, factoring (a+b+c), and then expanding the remaining determinant.

The derivation of the RHS is what I don't understand. I'm guessing that there is some elementary determinant operation at work that I'm unfamiliar with.

 [/INST] Add $\omega$ times the second row and $\omega^2$ times the third row to the first row to get
$\begin{align}
 \begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix} &= \begin{vmatrix}a+c\omega+b\omega^2 & b+a\omega+c\omega^2  & c+b\omega + a\omega^2 \\ c & a & b \\ b & c & a\end{vmatrix} \\
&=\begin{vmatrix}a+c\omega+b\omega^2 & \omega(a+b\omega^2+c\omega)  & \omega^2(c\omega+b\omega^2 + a) \\ c & a & b \\ b & c & a\end{vmatrix} 
\end{align}$
and hence $a+c\omega+b\omega^2$ is a factor. Similar arguments can be used to show the other two factors in the proof given by Lord Shark the Unkniwn.
</s>