We have,$
A=\left|\begin{array}{ccc}
1 & x & x+1 \\
2 x & x^2-x & x^2+x \\
3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right)
\end{array}\right|$
$\begin{aligned} & A=x^2(x-1)\left|\begin{array}{ccc}1 & x & x+1 \\ 2 & x-1 & x+1 \\ 3 & x-2 & x+1\end{array}\right| \\ & A=x(x-1)\left|\begin{array}{ccc}1 & x & x+1 \\ 0 & x-1-2 x & x+1-2 x-2 \\ 0 & x-2-3 x & x+1-3 x-3\end{array}\right| \\ & A=x(x-1)\left|\begin{array}{ccc}1 & x & x+1 \\ 0 & x^2-3 x & x^2-x-2 \\ 0 & -2 x-2 & -2 x-2\end{array}\right|\end{aligned}$
$\begin{gathered}A=x(x-1)(-2 x-2)\left|\begin{array}{ccc}1 & x & x+1 \\ 0 & x^2-3 x & x^2-x-2 \\ 0 & 1 & 1\end{array}\right| \\ A=-2 x(x-1)(x+1)\left(x^2-3 x-x^2+x+2\right) \\ A=-2 x(x-1)(x+1)(-2 x+2) \\ \quad A=4 x(x-1)^2(x+1)\end{gathered}$
$\therefore$ Rank of $A=2$ for all except$x=0,-1,1$
$\left({ }^*\right)$ No option is correct.