$\text{ The total number of distinct values for }x\in \mathrm{ℝ}\text{ of which }\left|\begin{array}{ccc}x& x^2& 1+x^3\\ 2x& 4x^2& 1+8x^3\\ 3x& 9x^2& 1+27x^3\end{array}\right|=10\text{ is }$

 $x^3\left|\begin{array}{ccc}1& 1& 1+x^3\\ 2& 4& 1+8x^3\\ 3& 9& 1+27x^3\end{array}\right|=10$

$\Rightarrow x^3\left(\right(4+108x^3-9-72x^3)-(2+54x^3-3-24x^3)+(1+x^3\left)\right(18-12\left)\right)=10$

$\Rightarrow x^3\left(\right(36x^3-5)-30x^3+1+6+6x^3)=10$

$\Rightarrow x^3(12x^3+2)=10$

$$\begin{gathered} \Rightarrow 6x^6+x^3-5=0 \\ \Rightarrow \left(x^3+1\right)\left(6x^3-5\right)=0 \\ \Rightarrow x=-1,{\left(\frac{5}{6}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} \end{gathered}$$

There are two distinct solutions are there for x