Given the equation:
$$\left(- 4 x + \left(x^{3} - 5 x^{2}\right)\right) + 20 = 0$$
transform
$$\left(- 4 x + \left(\left(- 5 x^{2} + \left(x^{3} - 8\right)\right) + 20\right)\right) + 8 = 0$$
or
$$\left(- 4 x + \left(\left(- 5 x^{2} + \left(x^{3} - 2^{3}\right)\right) + 5 \cdot 2^{2}\right)\right) + 2 \cdot 4 = 0$$
$$- 4 \left(x - 2\right) + \left(- 5 \left(x^{2} - 2^{2}\right) + \left(x^{3} - 2^{3}\right)\right) = 0$$
$$- 4 \left(x - 2\right) + \left(- 5 \left(x - 2\right) \left(x + 2\right) + \left(x - 2\right) \left(\left(x^{2} + 2 x\right) + 2^{2}\right)\right) = 0$$
Take common factor -2 + x from the equation
we get:
$$\left(x - 2\right) \left(\left(- 5 \left(x + 2\right) + \left(\left(x^{2} + 2 x\right) + 2^{2}\right)\right) - 4\right) = 0$$
or
$$\left(x - 2\right) \left(x^{2} - 3 x - 10\right) = 0$$
then:
$$x_{1} = 2$$
and also
we get the equation
$$x^{2} - 3 x - 10 = 0$$
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
$$a = 1$$
$$b = -3$$
$$c = -10$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (-10) = 49
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = 5$$
$$x_{3} = -2$$
The final answer for x^3 - 5*x^2 - 4*x + 20 = 0:
$$x_{1} = 2$$
$$x_{2} = 5$$
$$x_{3} = -2$$