Problem about degree of a polynomial and determinant

Question

answer is: $f(x)$ is polynomial of degree $2$

I couldn't proceed even the first step to find out the answer. please help

Answer 1

From a determinant, we can take values out from a row or a column and still end up with the same result. So, we can take $(1 + a^2)$ common from the first column, $(1 + b^2)$ common from the second column, and $(1 + c^2)$ common from the third column to arrive at:

\begin{align*} f(x) = (1 + a^2)(1 + b^2)(1 + c^2) \begin{vmatrix} \frac{1 + a^2x}{1 + a^2} & x & x \\ x & \frac{1 +b^2x}{1 + b^2} & x\\ x & x & \frac{1+c^2x}{1 + c^2} \end{vmatrix} \end{align*}

At this point, we can evaluate the determinant quite easily and notice that the answer will be a second degree polynomial