If $f(x) =\begin{vmatrix}a & -1 & 0\\ax & a & -1\\ax^2 & ax & a\end{vmatrix},$ then $f(2x)-f(x)$ equals

$ax(2a+3x)$

The correct answer is option (3) : $ax(2a+3x)$

Applying $R_2→R_2-xR_1 $ and $R_3→ R_3 -xR_2,$ we get

$f(c) = \begin{vmatrix}a & -1 & 0\\0 & a+x & -1\\0 & 0 & a+x\end{vmatrix}=a(a+x)^2$

$∴f(2x) -f(x) = a(a+ 2x)^2 -a(a+x)^2 = ax(2a+3x)$