(4-x)√4+3x-x²≤0 inequation

Given the inequality:
$$- x^{2} + \left(3 x + \sqrt{4} \left(4 - x\right)\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- x^{2} + \left(3 x + \sqrt{4} \left(4 - x\right)\right) = 0$$
Solve:
Expand the expression in the equation
$$- x^{2} + \left(3 x + \sqrt{4} \left(4 - x\right)\right) = 0$$
We get the quadratic equation
$$- x^{2} + x + 8 = 0$$
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
$$a = -1$$
$$b = 1$$
$$c = 8$$
, then

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

(1)^2 - 4 * (-1) * (8) = 33

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

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

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

or
$$x_{1} = \frac{1}{2} - \frac{\sqrt{33}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{33}}{2}$$
$$x_{1} = \frac{1}{2} - \frac{\sqrt{33}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{33}}{2}$$
$$x_{1} = \frac{1}{2} - \frac{\sqrt{33}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{33}}{2}$$
This roots
$$x_{1} = \frac{1}{2} - \frac{\sqrt{33}}{2}$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{33}}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{1}{2} - \frac{\sqrt{33}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{2}{5} - \frac{\sqrt{33}}{2}$$
substitute to the expression
$$- x^{2} + \left(3 x + \sqrt{4} \left(4 - x\right)\right) \leq 0$$
$$- \left(\frac{2}{5} - \frac{\sqrt{33}}{2}\right)^{2} + \left(3 \left(\frac{2}{5} - \frac{\sqrt{33}}{2}\right) + \sqrt{4} \left(4 - \left(\frac{2}{5} - \frac{\sqrt{33}}{2}\right)\right)\right) \leq 0$$

                 2              
     /      ____\      ____     
42   |2   \/ 33 |    \/ 33  <= 0
-- - |- - ------|  - ------     
5    \5     2   /      2        

one of the solutions of our inequality is:
$$x \leq \frac{1}{2} - \frac{\sqrt{33}}{2}$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \frac{1}{2} - \frac{\sqrt{33}}{2}$$
$$x \geq \frac{1}{2} + \frac{\sqrt{33}}{2}$$