$(x + 1).(x^{2} - x + 1) - 2x = x.(x + 1).(x - 1)$
$\Leftrightarrow x^{3} + 1 - 2x = x.(x^{2} - 1)$
$\Leftrightarrow x^{3} + 1 - 2x = x^{3} - x$
$\Leftrightarrow 1 - 2x = -x$
$\Leftrightarrow -2x + x = -1$
$\Leftrightarrow -x = -1$
$\Leftrightarrow x = 1$