Determinant of $\left(\begin{smallmatrix} -bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{smallmatrix}\right)$

Question

I started by adding $C_1$ goes to $C_1 + C_2 + C_3$.

After that well nothing seemed good to continue I have tried other steps too but it keeps getting more complex.

Answer 1

Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.

Thus, our determinant it's $$-a^2b^2c^2+2abc\prod_{cyc}(a+b)+\sum_{cyc}a^2b^2(a+c)(b+c)=$$ $$=-w^6+2w^3(9uv^2-w^3)+\sum_{cyc}a^2b^2(c^2+3v^2)=$$

$$=-w^6+2w^3(9uv^2-w^3)+3w^6+3v^2(9v^4-6uw^3)=27v^6=(ab+ac+bc)^3.$$