Explanation

(c)
LHS=$\left| \begin{matrix}
a + b & b + c & c + a \\
b + c & c + a & a + b \\
c + a & a + b & b + c
\end{matrix} \right|$
The determinant can be written sum of $2 \times 2 \times 2 = 8$ determinants of which 6 are reduces to zero because of their two rows are identical.
$\therefore\$ LHS=$2\left| \begin{array}{r}
a\ \ b\ \ c \\
b\ \ c\ \ a \\
c\ \ a\ \ b
\end{array} \right|$