Explanation

(b)
$f(x) = \left| \begin{matrix}
a & - 1 & 0 \\
ax & a & - 1 \\
ax^{2} & ax & a
\end{matrix} \right|$
Applying $R_{3} \rightarrow R_{3} - xR_{2}$
$f(x) = \left| \begin{matrix}
a & - 1 & 0 \\
ax & a & - 1 \\
0 & 0 & a + x
\end{matrix} \right| = (a + x)(a^{2} + ax)$
$\Rightarrow f(x) = a(a + x)^{2}$
$\therefore f(2x) = a(a + 2x)^{2}$
$\Rightarrow f(2x) - f(x) = ax(2a + 3x)$