The total number of distinct x ∈ R for which | cases x & x ^ 2 & 1...

{x}^{3}\left|\begin{cases}1 & 1 & 1 + {x}^{3} \\ 2 & 4 & 1 + 8{x}^{3} \\ 3 & 9 & 1 + 27{x}^{3}\end{cases}\right| = 10
$⇒$ {x}^{3}\left|\begin{cases}1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9\end{cases}\right| + {x}^{3} . {x}^{3}\left|\begin{cases}1 & 1 & 1 \\ 2 & 4 & 8 \\ 3 & 9 & 27\end{cases}\right| = 10
$⇒$ ${x}^{3}\left(25 - 23\right) + 6{x}^{6} . 2 = 10$
$⇒$ $6{x}^{6} + {x}^{3} - 5 = 0$
$⇒$ ${x}^{3} = \dfrac{5}{6}, - 1$
Two real solutions for x