Not able to factorize determinant

Question

I know how to solve this by determinant properties but I can't find any intuitive way to solve this by factorization method.

$$\Delta=\left|\begin{matrix}b^2+c^2&ab&ac\\ab&c^2+a^2&bc\\ca&cb&a^2+b^2\end{matrix}\right|$$

Answer 1

If $c=0$,

$$\Delta=\left|\begin{matrix}b^2&ab&0\\ab&a^2&0\\0&0&a^2+b^2\end{matrix}\right|=0$$

hence by symmetry $abc$ must be a factor. Then if you look at the terms implied in Sarrus' rule, they all include the variables squared. Hence $\lambda a^2b^2c^2$, and

$$\lambda=\left|\begin{matrix}2&1&1\\1&2&1\\1&1&2\end{matrix}\right|.$$