(x^2)-(4*x)-2>0 inequation

Given the inequality:
$$\left(x^{2} - 4 x\right) - 2 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x^{2} - 4 x\right) - 2 = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = -2$$
, then

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

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

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

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

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

or
$$x_{1} = 2 + \sqrt{6}$$
$$x_{2} = 2 - \sqrt{6}$$
$$x_{1} = 2 + \sqrt{6}$$
$$x_{2} = 2 - \sqrt{6}$$
$$x_{1} = 2 + \sqrt{6}$$
$$x_{2} = 2 - \sqrt{6}$$
This roots
$$x_{2} = 2 - \sqrt{6}$$
$$x_{1} = 2 + \sqrt{6}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$\left(2 - \sqrt{6}\right) + - \frac{1}{10}$$
=
$$\frac{19}{10} - \sqrt{6}$$
substitute to the expression
$$\left(x^{2} - 4 x\right) - 2 > 0$$
$$-2 + \left(\left(\frac{19}{10} - \sqrt{6}\right)^{2} - 4 \left(\frac{19}{10} - \sqrt{6}\right)\right) > 0$$

                   2              
  48   /19     ___\        ___    
- -- + |-- - \/ 6 |  + 4*\/ 6  > 0
  5    \10        /               
    

one of the solutions of our inequality is:
$$x < 2 - \sqrt{6}$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 - \sqrt{6}$$
$$x > 2 + \sqrt{6}$$