If $a^2+b^2+c^2=-2$  and  $f\left(x\right)=\left|\begin{array}{ccc}1+a^{2}x& \left(1+b^2\right)x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& 1+b^{2}x& \left(1+c^2\right)x\\ \left(1+a^2\right)x& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|$then$f\left(x\right)$ is a polynomial of degree

Applying , $C_1\to C_1+C_2+C_3$ we get 

$f\left(x\right)=\left|\begin{array}{ccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x\\ 1& 1+b^{2}x& \left(1+c^2\right)x\\ 1& \left(1+b^2\right)x& 1+c^{2}x\end{array}\right|$

                       [using $a^2+b^2+c^2=-2$]

Applying $R_2\to R_2-R_1$and $R_3\to R_3-R_1,$we get 

$f\left(x\right)=\left|\begin{array}{cccc}1& \left(1+b^2\right)x& \left(1+c^2\right)x& \\ 0& 1& x& 0\\ 0& 0& 1-x& \end{array}\right|=(1-x{)}^2$

which is a polynomial of degree 2.