<s>[INST] Help Factoring A Difficult Expression Given  A = $\begin{bmatrix}a & b & c \\c & a & b \\b & c & a \\\end{bmatrix}$, B =$\begin{bmatrix}a & b & c \\a^2 & b^2 & c^2 \\a^4 & b^4 & c^4 \\\end{bmatrix}$

Suppose $C =AB=\begin{bmatrix}
a^2 + a^2b + a^4c & ab + b^3 + b^4c & ac +bc^2 + c^5 \\
\\
ca + a^3 +a^4b & cb + ab^2 + b^5 & c^2 + ac^2 +bc^4 \\
\\
ba + a^2c + a^5 & b^2 + b^2c + ab^4 & bc + c^3 + ac^4 
\end{bmatrix}$ 

I have been asked to show that the following holds:

$det(C)= -abc(b-c)(a-c)(a-b)(a^2-ab+b^2 -bc +c^2 -ac)(a+b+c)^2$

The way i proceeded to go about this is using the fact that $det(AB) = det(A)det(B)$.Now finding the determinant of A and B i have the following.

$det(A)= a^3 -3abc +b^3 +c^3$

$det(B) = abc(c^3b - cb^3 - c^3a + ca^3 + b^3a - ba^3)$

The problem i have is factoring out the necessary factors from the product of the determinants. I have tried loads of methods now and i have got nowhere and am a bit depressed by it. Any ideas would be really appreciated.
Best Regards
 [/INST] Hint: $$a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
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