(x^2+9x+18)(x^2+4x+5)<=0 inequation

Given the inequality:
$$\left(\left(x^{2} + 4 x\right) + 5\right) \left(\left(x^{2} + 9 x\right) + 18\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(x^{2} + 4 x\right) + 5\right) \left(\left(x^{2} + 9 x\right) + 18\right) = 0$$
Solve:
Given the equation:
$$\left(\left(x^{2} + 4 x\right) + 5\right) \left(\left(x^{2} + 9 x\right) + 18\right) = 0$$
Because the right side of the equation is zero, then the solution of the equation is exists if at least one of the multipliers in the left side of the equation equal to zero.
We get the equations
$$x^{2} + 4 x + 5 = 0$$
$$x^{2} + 9 x + 18 = 0$$
solve the resulting equation:
1.
$$x^{2} + 4 x + 5 = 0$$
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
$$a = 1$$
$$b = 4$$
$$c = 5$$
, then

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

(4)^2 - 4 * (1) * (5) = -4

Because D<0, then the equation
has no real roots,
but complex roots is exists.

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

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

or
$$x_{1} = -2 + i$$
$$x_{2} = -2 - i$$
2.
$$x^{2} + 9 x + 18 = 0$$
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:
$$x_{3} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{4} = \frac{- \sqrt{D} - b}{2 a}$$
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) = 9

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

x3 = (-b + sqrt(D)) / (2*a)

x4 = (-b - sqrt(D)) / (2*a)

or
$$x_{3} = -3$$
$$x_{4} = -6$$
$$x_{1} = -2 + i$$
$$x_{2} = -2 - i$$
$$x_{3} = -3$$
$$x_{4} = -6$$
Exclude the complex solutions:
$$x_{1} = -3$$
$$x_{2} = -6$$
This roots
$$x_{2} = -6$$
$$x_{1} = -3$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-6 + - \frac{1}{10}$$
=
$$- \frac{61}{10}$$
substitute to the expression
$$\left(\left(x^{2} + 4 x\right) + 5\right) \left(\left(x^{2} + 9 x\right) + 18\right) \leq 0$$
$$\left(5 + \left(\frac{\left(-61\right) 4}{10} + \left(- \frac{61}{10}\right)^{2}\right)\right) \left(\left(\frac{\left(-61\right) 9}{10} + \left(- \frac{61}{10}\right)^{2}\right) + 18\right) \leq 0$$

55211     
----- <= 0
10000     

but

55211     
----- >= 0
10000     

Then
$$x \leq -6$$
no execute
one of the solutions of our inequality is:
$$x \geq -6 \wedge x \leq -3$$

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       x2      x1