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.

Let $$a=b=c=1$$ and you get the matrix 
$$\begin{vmatrix}4&1&1\\1 &4&1 \\1&1& 4\end{vmatrix}=27k$$
The determinant is easily evaluated to be $54$ so $$27k=54$$.
Thus $$k=2$$ 
