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Finding a rational solution to a monic polynomial equation

I have an equation, and I am trying to find an integer solution to it. Since I am not finding a rational solution, is this the right way to approach this problem?
Am I trying to assume that it is 0/0?
Edit: I am trying to solve the following equation: $x^3 + x^2 – 3x – 1 = 0$
The answer I am getting is $x = 1$ or $x = \pm 3$

A:

$$x^3 + x^2 – 3x – 1 = (x-1)(x^2+x+1) = (x-1)(x-2)(x+1)$$
$$\implies (x-1)^2 (x+1)=0$$
You’re looking for integers $x$ such that $x+1=0$, hence $x=-1$