x+6/(5x+10)(x-5)>0 inequation

Given the inequality:
$$x + \left(x - 5\right) \frac{6}{5 x + 10} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x + \left(x - 5\right) \frac{6}{5 x + 10} = 0$$
Solve:
Given the equation:
$$x + \left(x - 5\right) \frac{6}{5 x + 10} = 0$$
Multiply the equation sides by the denominators:
10 + 5*x
we get:
$$\left(x + \left(x - 5\right) \frac{6}{5 x + 10}\right) \left(5 x + 10\right) = 0$$
$$5 x^{2} + 16 x - 30 = 0$$
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
$$a = 5$$
$$b = 16$$
$$c = -30$$
, then

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

(16)^2 - 4 * (5) * (-30) = 856

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

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

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

or
$$x_{1} = - \frac{8}{5} + \frac{\sqrt{214}}{5}$$
$$x_{2} = - \frac{\sqrt{214}}{5} - \frac{8}{5}$$
$$x_{1} = - \frac{8}{5} + \frac{\sqrt{214}}{5}$$
$$x_{2} = - \frac{\sqrt{214}}{5} - \frac{8}{5}$$
$$x_{1} = - \frac{8}{5} + \frac{\sqrt{214}}{5}$$
$$x_{2} = - \frac{\sqrt{214}}{5} - \frac{8}{5}$$
This roots
$$x_{2} = - \frac{\sqrt{214}}{5} - \frac{8}{5}$$
$$x_{1} = - \frac{8}{5} + \frac{\sqrt{214}}{5}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(- \frac{\sqrt{214}}{5} - \frac{8}{5}\right) + - \frac{1}{10}$$
=
$$- \frac{\sqrt{214}}{5} - \frac{17}{10}$$
substitute to the expression
$$x + \left(x - 5\right) \frac{6}{5 x + 10} > 0$$
$$\left(- \frac{\sqrt{214}}{5} - \frac{17}{10}\right) + \left(-5 + \left(- \frac{\sqrt{214}}{5} - \frac{17}{10}\right)\right) \frac{6}{5 \left(- \frac{\sqrt{214}}{5} - \frac{17}{10}\right) + 10} > 0$$

                   /         _____\    
                   |  67   \/ 214 |    
         _____   6*|- -- - -------|    
  17   \/ 214      \  10      5   /    
- -- - ------- + ------------------ > 0
  10      5         3     _____        
                    - - \/ 214         
                    2                  
    

Then
$$x < - \frac{\sqrt{214}}{5} - \frac{8}{5}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\sqrt{214}}{5} - \frac{8}{5} \wedge x < - \frac{8}{5} + \frac{\sqrt{214}}{5}$$

         _____  
        /     \  
-------ο-------ο-------
       x2      x1