Explanation

(b)
$\left| \begin{matrix}
x & p & q \\
p & x & q \\
p & q & x
\end{matrix} \right| = \left| \begin{array}{r}
x + p + q\ \ \ \ p\ \ \ \ q \\
x + p + q\ \ \ \ x\ \ \ \ q \\
x + p + q\ \ \ \ q\ \ \ \ x
\end{array} \right|$
$= (x + p + q)\left| \begin{array}{r}
1\ \ p\ \ q \\
1\ \ x\ \ q \\
1\ \ q\ \ x
\end{array} \right|$
$= (x + p + q)\left| \begin{matrix}
1 & p & q \\
0 & x - p & 0 \\
0 & q - p & x - q
\end{matrix} \right|$
$= (x + p + q)\begin{bmatrix}
x - p & 0 \\
q - p & x - q
\end{bmatrix}$
$= (x - p)(x - q)(x + p + q)$