Explanation

(d)
$\left| \begin{matrix}
4 + x^{2} & - 6 & - 2 \\
- 6 & 9 + x^{2} & 3 \\
- 2 & 3 & 1 + x^{2}
\end{matrix} \right|$
$= \left( 4 + x^{2} \right)\lbrack\left( 1 + x^{2} \right)\left( 9 + x^{2} \right) - 9\rbrack$
$+ 6\left\lbrack - 6\left( 1 + x^{2} \right) + 6 \right\rbrack - 2\lbrack - 18 + 2(9 + x^{2})\rbrack$
$= \left( 4 + x^{2} \right)\left( 10x^{2} + x^{4} \right) - 36x^{2} - 4x^{2}$
$= 40x^{2} + 4x^{4} + 10x^{4} + x^{6} - 40x^{2}$
$= x^{4}\left( x^{2} + 14 \right)$
Which is not divisible by $x^{5}.$