(x+2)9-x^2>0 inequation

Given the inequality:
$$- x^{2} + 9 \left(x + 2\right) > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$- x^{2} + 9 \left(x + 2\right) = 0$$
Solve:
Expand the expression in the equation
$$- x^{2} + 9 \left(x + 2\right) = 0$$
We get the quadratic equation
$$- x^{2} + 9 x + 18 = 0$$
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
$$a = -1$$
$$b = 9$$
$$c = 18$$
, then

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

(9)^2 - 4 * (-1) * (18) = 153

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

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

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

or
$$x_{1} = \frac{9}{2} - \frac{3 \sqrt{17}}{2}$$
$$x_{2} = \frac{9}{2} + \frac{3 \sqrt{17}}{2}$$
$$x_{1} = \frac{9}{2} - \frac{3 \sqrt{17}}{2}$$
$$x_{2} = \frac{9}{2} + \frac{3 \sqrt{17}}{2}$$
$$x_{1} = \frac{9}{2} - \frac{3 \sqrt{17}}{2}$$
$$x_{2} = \frac{9}{2} + \frac{3 \sqrt{17}}{2}$$
This roots
$$x_{1} = \frac{9}{2} - \frac{3 \sqrt{17}}{2}$$
$$x_{2} = \frac{9}{2} + \frac{3 \sqrt{17}}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{9}{2} - \frac{3 \sqrt{17}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{22}{5} - \frac{3 \sqrt{17}}{2}$$
substitute to the expression
$$- x^{2} + 9 \left(x + 2\right) > 0$$
$$- \left(\frac{22}{5} - \frac{3 \sqrt{17}}{2}\right)^{2} + 9 \left(\left(\frac{22}{5} - \frac{3 \sqrt{17}}{2}\right) + 2\right) > 0$$

                     2                
      /         ____\         ____    
288   |22   3*\/ 17 |    27*\/ 17  > 0
--- - |-- - --------|  - ---------    
 5    \5       2    /        2        

Then
$$x < \frac{9}{2} - \frac{3 \sqrt{17}}{2}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{9}{2} - \frac{3 \sqrt{17}}{2} \wedge x < \frac{9}{2} + \frac{3 \sqrt{17}}{2}$$

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