Explanation

(c)
$\left| \begin{matrix}
b^{2} + c^{2} & a^{2} & a^{2} \\
b^{2} & c^{2} + a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2} + b^{2}
\end{matrix} \right|$
$= - 2\left| \begin{matrix}
0 & c^{2} & b^{2} \\
b^{2} & c^{2} + a^{2} & b^{2} \\
c^{2} & c^{2} & a^{2} + b^{2}
\end{matrix} \right|$ $\ \lbrack R_{1} \rightarrow R_{1} - \left( R_{2} + R_{3} \right)\rbrack$
$= - 2\left| \begin{matrix}
0 & c^{2} & b^{2} \\
b^{2} & a^{2} & 0 \\
c^{2} & 0 & a^{2}
\end{matrix} \right|$ $\ \binom{R_{2} \rightarrow R_{2} - R_{1}}{R_{3} \rightarrow R_{3} - R_{1}}$
$= - 2\left\lbrack - c^{2}\left( b^{2}a^{2} - 0 \right) + b^{2}\left( 0 - a^{2}c^{2} \right) \right\rbrack$
$= - 2\left\lbrack - {2a}^{2}b^{2}c^{2} \right\rbrack = 4a^{2}b^{2}c^{2}$