$ x^3-3x+2=0 $

Adam, Paul and Alek solved the equation $$ x^3-3x+2=0 $$ each in their own way.

Adam factored the polynomial into a product of polynomials of smaller degree and solved the equation as follows: $$ \begin{gather} x^2\left(x+2\right)-2x\left(x+2\right)+x+2=0 \cr \left(x+2\right)\left(x^2-2x\right)=0 \cr x\left(x+2\right)\left(x-2\right)=0 \cr x_1=0,\ x_2=-2,\ x_3=2 \end{gather} $$

Alek factored the polynomial and solved the equation this way: $$ \begin{gather} x\left(x^2-1\right)-2\left(x-1\right)=0 \cr x\left(x-1\right)\left(x+1\right)-2\left(x-1\right)=0 \cr \left(x-1\right)\left(x^2+x-2\right)=0 \cr x_1=1 \cr x_{2,3}=\frac{-1\pm\sqrt{1^2-4\cdot 1\cdot \left(-2\right)}}{2\cdot 1} \cr x_2=1,\ x_3=-2 \end{gather} $$ As for him, the equation has the roots: $x_1=x_2=1$ (a double root) and $x_3=-2$.

Paul guessed a root $x_1=-2$ and found the remaining roots $x_2$ and $x_3$ by the formula: $$ \begin{gather} x_{2,3}=\frac{-\left(-3\right)\pm\sqrt{\left(-3\right)^2-4\cdot 1\cdot 2}}{2\cdot1} \cr x_2=1,\ x_3=2 \end{gather} $$ Which one of them proceeded correctly in solving?