x+12/(x-14)(7x+2)<=0 inequation

Given the inequality:
$$x + \frac{12}{x - 14} \left(7 x + 2\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$x + \frac{12}{x - 14} \left(7 x + 2\right) = 0$$
Solve:
Given the equation:
$$x + \frac{12}{x - 14} \left(7 x + 2\right) = 0$$
Multiply the equation sides by the denominators:
-14 + x
we get:
$$\left(x - 14\right) \left(x + \frac{12}{x - 14} \left(7 x + 2\right)\right) = 0$$
$$x^{2} + 70 x + 24 = 0$$
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
$$a = 1$$
$$b = 70$$
$$c = 24$$
, then

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

(70)^2 - 4 * (1) * (24) = 4804

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

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

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

or
$$x_{1} = -35 + \sqrt{1201}$$
$$x_{2} = -35 - \sqrt{1201}$$
$$x_{1} = -35 + \sqrt{1201}$$
$$x_{2} = -35 - \sqrt{1201}$$
$$x_{1} = -35 + \sqrt{1201}$$
$$x_{2} = -35 - \sqrt{1201}$$
This roots
$$x_{2} = -35 - \sqrt{1201}$$
$$x_{1} = -35 + \sqrt{1201}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(-35 - \sqrt{1201}\right) + - \frac{1}{10}$$
=
$$- \frac{351}{10} - \sqrt{1201}$$
substitute to the expression
$$x + \frac{12}{x - 14} \left(7 x + 2\right) \leq 0$$
$$\left(- \frac{351}{10} - \sqrt{1201}\right) + \frac{12}{\left(- \frac{351}{10} - \sqrt{1201}\right) - 14} \left(7 \left(- \frac{351}{10} - \sqrt{1201}\right) + 2\right) \leq 0$$

                      /  2437       ______\     
                   12*|- ---- - 7*\/ 1201 |     
  351     ______      \   10              /     
- --- - \/ 1201  + ------------------------ <= 0
   10                    491     ______         
                       - --- - \/ 1201          
                          10                    

one of the solutions of our inequality is:
$$x \leq -35 - \sqrt{1201}$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq -35 - \sqrt{1201}$$
$$x \geq -35 + \sqrt{1201}$$