Explanation

(c)
$\mathrm{\Delta} = \left| \begin{matrix}
1 & 1 & 1 \\
bc & ca & ab \\
b + c & c + a & a + b
\end{matrix} \right|$
$= \left| \begin{matrix}
1 & 0 & 0 \\
bc & c(a - b) & a(b - c) \\
b + c & (a - b) & (b - c)
\end{matrix} \right|\left\lbrack \begin{array}{r}
C_{2}{\rightarrow C}_{2} - C_{1} \\
C_{3}{\rightarrow C}_{3} - C_{2}
\end{array} \right\rbrack$
$= (a - b)(b - c)\ \left| \begin{matrix}
1 & 0 & 0 \\
bc & c & a \\
b + c & 1 & 1
\end{matrix} \right|$
$= (a - b)(b - c)(c - a)$