The determinant $\left|\begin{array}{ccc}4 + x^2& -6& -2\\ -6& 9+x^2& 3\\ -2& 3& 1+x^2\end{array}\right|$ is not divisible by

$\begin{array}{rcl} & & (\mathrm{x}+{\mathrm{x}}^2) (9+{\mathrm{x}}^2) (1+{\mathrm{x}}^2)-(4+{\mathrm{x}}^2)9+6(-6(1+{\mathrm{x}}^2)+6)-2(-18+2(9+{\mathrm{x}}^2\left)\right)\\ & =& (4+{\mathrm{x}}^2) (9+{\mathrm{x}}^2) (1+{\mathrm{x}}^2)-9(4+{\mathrm{x}}^2)-36{\mathrm{x}}^2-4{\mathrm{x}}^2\\ & =& (4+{\mathrm{x}}^2) (9+{\mathrm{x}}^2) (1+{\mathrm{x}}^2)-9(4+{\mathrm{x}}^2)-40{\mathrm{x}}^2\\ & =& (4+{\mathrm{x}}^2) \left[9+9{\mathrm{x}}^2+{\mathrm{x}}^2+{\mathrm{x}}^4-9\right]-40{\mathrm{x}}^2\\ & =& (4+{\mathrm{x}}^2) (10{\mathrm{x}}^2+{\mathrm{x}}^4)-40{\mathrm{x}}^2\\ & =& {\mathrm{x}}^2\left[(4+{\mathrm{x}}^2) (10+{\mathrm{x}}^2)-40\right]\end{array}$

$\therefore$ it is not divisible by ${\mathrm{x}}^5$