Can I use this algorithm for solving cubic equations?

Question

I am trying to find the root solutions for a cubic equation including the eigenvalues of each root.

I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has eigenvalues of $2$. for instance the following equation : $-r^3+3r-2=0$.

I found an algorithm for finding the roots and eignevalues for this equation:

1. finding the parameters $a$ and $d$ ($a$ is the coefficient of $r^3$ and $d$ is the coefficient of the integer"

2. write the factors of $a$ and $d$ in $2$ different lists

3. assign each value to the equation and see if the equation has a solution.

if there is a solution - so the factor is a solution root for the equation.

for illustration: for the following equation of before $-r^3+3r-2=0$

$a= -1$ and the factors of -1 are $\left[ 1,-1 \right] $

$d=-2$ and the factors of $-2$ are $\left[ 1,-1,2,-2 \right] $

assigning each factor separately tells us that -1 is a solution root with eigenvalue of $2$ and $-2$ is a solution root with eigenvalue of $1$.

my question is if this algorithm is correct or if there is any other algorithm that works and not too long?

thank you