Find the constant $k$ from the determinant 
Given:  $$\begin{vmatrix}(b+c)^2 &a^2&a^2\\b^2 &(c+a)^2&b^2 \\c^2&c^2& (a+b)^2\end{vmatrix}=k(abc)(a+b+c)^3$$ Find $k$.

If I directly open the determinant it will go to long I can't apply most of the row or column operation as they keep making it more complex.

It's not long:
$$\prod_{cyc}(a+b)^2+2a^2b^2c^2-\sum_{cyc}a^2b^2(a+b)^2=$$
$$\sum_{cyc}\left(a^4b^2+a^4c^2+2a^3b^3+2a^4bc+2a^3b^2c+2a^3c^2b+2a^2b^2c^2+4a^3b^2c+4a^3c^2b+\tfrac{4}{3}a^2b^2c^2\right)+$$
$$+\sum_{cyc}\left(\frac{2}{3}a^2b^2c^2-a^4b^2-a^4c^2-2a^3b^3\right)=$$
$$=\sum_{cyc}(2a^4bc+6a^3b^2c+6a^3c^2b+4a^2b^2c^2)=2abc(a+b+c)^3.$$
Two minutes of computations! By hand, of course. 
