(sinx+5)(sqrt((x+3)^2)-4)<=0 inequation

Given the inequality:
$$\left(\sqrt{\left(x + 3\right)^{2}} - 4\right) \left(\sin{\left(x \right)} + 5\right) \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\sqrt{\left(x + 3\right)^{2}} - 4\right) \left(\sin{\left(x \right)} + 5\right) = 0$$
Solve:
$$x_{1} = -7$$
$$x_{2} = 1$$
$$x_{3} = \pi + \operatorname{asin}{\left(5 \right)}$$
$$x_{4} = - \operatorname{asin}{\left(5 \right)}$$
Exclude the complex solutions:
$$x_{1} = -7$$
$$x_{2} = 1$$
This roots
$$x_{1} = -7$$
$$x_{2} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-7 + - \frac{1}{10}$$
=
$$- \frac{71}{10}$$
substitute to the expression
$$\left(\sqrt{\left(x + 3\right)^{2}} - 4\right) \left(\sin{\left(x \right)} + 5\right) \leq 0$$
$$\left(-4 + \sqrt{\left(- \frac{71}{10} + 3\right)^{2}}\right) \left(\sin{\left(- \frac{71}{10} \right)} + 5\right) \leq 0$$

       /71\     
    sin|--|     
1      \10/ <= 0
- - -------     
2      10       

but

       /71\     
    sin|--|     
1      \10/ >= 0
- - -------     
2      10       

Then
$$x \leq -7$$
no execute
one of the solutions of our inequality is:
$$x \geq -7 \wedge x \leq 1$$

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