Explanation

(d)
Let $\mathrm{\Delta} = \left| \begin{matrix}
a\ ^{2} & b^{2} & c^{2} \\
(a + 1)^{2} & (b + 1)^{2} & (c + 1)^{2} \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
Applying $R_{2} \rightarrow R_{2} - R_{3}$
$= \left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
4a & 4b & 4c \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
$= 4\left| \begin{matrix}
a^{2} & b^{2} & c^{2} \\
a & b & c \\
(a - 1)^{2} & (b - 1)^{2} & (c - 1)^{2}
\end{matrix} \right|$
Applying $R_{3} \rightarrow R_{3} - {(R}_{1} - 2R_{2})$
=4$\left| \begin{array}{r}
a^{2}\ \ b^{2}\ \ c^{2} \\
a\ \ \ \ \ b\ \ \ \ \ c \\
1\ \ \ \ \ \ 1\ \ \ \ \ 1
\end{array} \right|$
$\therefore k = 4$