x^3-3*x-2=0 equation

Given the equation:
$$\left(x^{3} - 3 x\right) - 2 = 0$$
transform
$$\left(- 3 x + \left(x^{3} + 1\right)\right) - 3 = 0$$
or
$$\left(- 3 x + \left(x^{3} - \left(-1\right)^{3}\right)\right) - 3 = 0$$
$$- 3 \left(x + 1\right) + \left(x^{3} - \left(-1\right)^{3}\right) = 0$$
$$\left(x + 1\right) \left(\left(x^{2} - x\right) + \left(-1\right)^{2}\right) - 3 \left(x + 1\right) = 0$$
Take common factor 1 + x from the equation
we get:
$$\left(x + 1\right) \left(\left(\left(x^{2} - x\right) + \left(-1\right)^{2}\right) - 3\right) = 0$$
or
$$\left(x + 1\right) \left(x^{2} - x - 2\right) = 0$$
then:
$$x_{1} = -1$$
and also
we get the equation
$$x^{2} - x - 2 = 0$$
This equation is of the form

a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = -2$$
, then

D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1) * (-2) = 9

Because D > 0, then the equation has two roots.

x2 = (-b + sqrt(D)) / (2*a)

x3 = (-b - sqrt(D)) / (2*a)

or
$$x_{2} = 2$$
$$x_{3} = -1$$
The final answer for x^3 - 3*x - 2 = 0:
$$x_{1} = -1$$
$$x_{2} = 2$$
$$x_{3} = -1$$