Explanation

(d)
Let $\mathrm{\Delta} = \left| \begin{matrix}
a + b + 2c & a & b \\
c & 2a + b + c & b \\
c & a & a + 2b + c
\end{matrix} \right|$
Applying $C_{1} \rightarrow C_{1} + C_{2} + C_{3}$
$= \left| \begin{matrix}
2(a + b + c) & a & b \\
2(a + b + c) & 2a + b + c & b \\
2(a + b + c) & a & a + 2b + c
\end{matrix} \right|$
$= 2(a + b + c)\left| \begin{matrix}
1 & a & b \\
1 & 2a + b + c & b \\
1 & a & a + 2b + c
\end{matrix} \right|$
=$2(a + b + c)\left| \begin{matrix}
0 & - (a + b + c) & 0 \\
0 & (a + b + c) & - (a + b + c) \\
1 & a & a + 2b + c
\end{matrix} \right|$ $\binom{R_{1} \rightarrow R_{1} - R_{2}}{R_{2} \rightarrow R_{2} - R_{3}}$
$= 2(a + b + c)^{3}\left| \begin{matrix}
0 & - 1 & 0 \\
0 & 1 & - 1 \\
1 & a & a + 2b + c
\end{matrix} \right|$
$= 2(a + b + c)^{3}$