x*(x+1)*(x+2)*(x+3)=0,5625 equation

Given the equation:
$$x \left(x + 1\right) \left(x + 2\right) \left(x + 3\right) = 0.5625$$
transform:
Take common factor from the equation
$$6.75 \left(0.666666666666667 x + 1\right)^{2} \left(0.333333333333333 x^{2} + 1 x - 0.0833333333333333\right) = 0$$
Because the right side of the equation is zero, then the solution of the equation is exists if at least one of the multipliers in the left side of the equation equal to zero.
We get the equations
$$2.25 x^{2} + 6.75 x - 0.5625 = 0$$
$$0.666666666666667 x + 1 = 0$$
solve the resulting equation:
1.
$$2.25 x^{2} + 6.75 x - 0.5625 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 2.25$$
$$b = 6.75$$
$$c = -0.5625$$
, then

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

(6.75)^2 - 4 * (2.25) * (-0.5625) = 50.625

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

x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = 0.0811388300841897$$
$$x_{2} = -3.08113883008419$$
2.
$$0.666666666666667 x + 1 = 0$$
Move free summands (without x)
from left part to right part, we given:
$$0.666666666666667 x = -1$$
Divide both parts of the equation by 0.666666666666667

x = -1 / (0.666666666666667)

We get the answer: x3 = -1.5
The final answer:
$$x_{1} = 0.0811388300841897$$
$$x_{2} = -3.08113883008419$$
$$x_{3} = -1.5$$