-5x^2-11*x-6>=0 inequation

Given the inequality:
$$\left(- 5 x^{2} - 11 x\right) - 6 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(- 5 x^{2} - 11 x\right) - 6 = 0$$
Solve:
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
$$a = -5$$
$$b = -11$$
$$c = -6$$
, then

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

(-11)^2 - 4 * (-5) * (-6) = 1

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

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

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

or
$$x_{1} = - \frac{6}{5}$$
$$x_{2} = -1$$
$$x_{1} = - \frac{6}{5}$$
$$x_{2} = -1$$
$$x_{1} = - \frac{6}{5}$$
$$x_{2} = -1$$
This roots
$$x_{1} = - \frac{6}{5}$$
$$x_{2} = -1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{6}{5} + - \frac{1}{10}$$
=
$$- \frac{13}{10}$$
substitute to the expression
$$\left(- 5 x^{2} - 11 x\right) - 6 \geq 0$$
$$-6 + \left(- 5 \left(- \frac{13}{10}\right)^{2} - \frac{\left(-13\right) 11}{10}\right) \geq 0$$

-3/20 >= 0

but

-3/20 < 0

Then
$$x \leq - \frac{6}{5}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{6}{5} \wedge x \leq -1$$

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        /     \  
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       x1      x2