Find the value of a third order circulant type determinant

Question

To calculate the value of the determinant\begin{pmatrix} (b+c)^2 &c^2 &b^2 \\ c^2 &(c+a)^2 &a^2 \\ b^2 &a^2&(a+b)^2 \end{pmatrix} . I multiplied R1,R2,R3 by a²,b²,c² respectively and then used the operations R1=R1-R2,R2=R2-R3,R3=R3-R1 which got me a zero valued determinant eventually. The correct answer given is 2(bc+ca+ab)². What's wrong with the row Operations that I used? Is there any other suitable Operations?

Answer 1

It's $$\prod_{cyc}(a+b)^2+2a^2b^2c^2-\sum_{cyc}a^4(b+c)^2=$$ $$=2\sum_{cyc}(a^3b^3+3a^3b^2c+3a^3c^2b+2a^2b^2c^2)=2(ab+ac+bc)^3.$$

I used the following.

$$\prod_{cyc}(a+b)=\sum_{cyc}\left(a^2b+a^2c+\frac{2}{3}abc\right),$$ $$\left(\sum_{cyc}(a^2b+a^2c)\right)^2=\sum_{cyc}(a^4b^2+a^4c^2+2a^3b^3+2a^4bc+2a^3b^2c+2a^3c^2b+2a^2b^2c^2)$$ and $$\sum_{cyc}a^4(b+c)^2=\sum_{cyc}(a^4b^2+a^4c^2+2a^4bc).$$