(2x-1)(3x+6)≥0 inequation

Given the inequality:
$$\left(2 x - 1\right) \left(3 x + 6\right) \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 x - 1\right) \left(3 x + 6\right) = 0$$
Solve:
Expand the expression in the equation
$$\left(2 x - 1\right) \left(3 x + 6\right) = 0$$
We get the quadratic equation
$$6 x^{2} + 9 x - 6 = 0$$
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
$$a = 6$$
$$b = 9$$
$$c = -6$$
, then

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

(9)^2 - 4 * (6) * (-6) = 225

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

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

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

or
$$x_{1} = \frac{1}{2}$$
$$x_{2} = -2$$
$$x_{1} = \frac{1}{2}$$
$$x_{2} = -2$$
$$x_{1} = \frac{1}{2}$$
$$x_{2} = -2$$
This roots
$$x_{2} = -2$$
$$x_{1} = \frac{1}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left(2 x - 1\right) \left(3 x + 6\right) \geq 0$$
$$\left(\frac{\left(-21\right) 2}{10} - 1\right) \left(\frac{\left(-21\right) 3}{10} + 6\right) \geq 0$$

39     
-- >= 0
25     

one of the solutions of our inequality is:
$$x \leq -2$$

 _____           _____          
      \         /
-------•-------•-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -2$$
$$x \geq \frac{1}{2}$$