Explanation

(d)
We have, $\mathrm{\Delta} = \frac{1}{abc}\left| \begin{matrix}
a^{3} + ax & a^{2}b & a^{2}c \\
ab^{2} & b^{3} + bx & b^{2}c \\
c^{2}a & c^{2}b & c^{2} + xc
\end{matrix} \right|$
Taking $a,\ b,c$ common in columns Ist, IInd and IIIrd, we get,
$\mathrm{\Delta} = \left| \begin{matrix}
a^{2} + x & a^{2} & a^{2} \\
b^{2} & b^{2} + x & b^{2} \\
c^{2} & c^{2} & c^{2} + x
\end{matrix} \right|$
Applying $R_{1} \rightarrow R_{1} + R_{2} + R_{3}$
$= (a^{2} + b^{2} + c^{2} + x)\left| \begin{matrix}
1 & 1 & 1 \\
b^{2} & b^{2} + x & b^{2} \\
c^{2} & c^{2} & c^{2} + x
\end{matrix} \right|$
Applying $C_{2} \rightarrow C_{2} - C_{1},\ C_{3} \rightarrow C_{3} - C_{1}$
$= (a^{2} + b^{2} + c^{2} + x)\left| \begin{matrix}
1 & 1 & 0 \\
b^{2} & x & 0 \\
c^{2} & 0 & x
\end{matrix} \right|$
$= x\left( x - b^{2} \right)(a^{2} + b^{2} + c^{2} + x)$
Hence, option (d) is correct.