(x+1)*log(x+4)<0 inequation

Given the inequality:
$$\left(x + 1\right) \log{\left(x + 4 \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x + 1\right) \log{\left(x + 4 \right)} = 0$$
Solve:
$$x_{1} = -3$$
$$x_{2} = -1$$
$$x_{1} = -3$$
$$x_{2} = -1$$
This roots
$$x_{1} = -3$$
$$x_{2} = -1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-3 + - \frac{1}{10}$$
=
$$- \frac{31}{10}$$
substitute to the expression
$$\left(x + 1\right) \log{\left(x + 4 \right)} < 0$$
$$\left(- \frac{31}{10} + 1\right) \log{\left(- \frac{31}{10} + 4 \right)} < 0$$

-21*log(9/10)    
------------- < 0
      10         

but

-21*log(9/10)    
------------- > 0
      10         

Then
$$x < -3$$
no execute
one of the solutions of our inequality is:
$$x > -3 \wedge x < -1$$

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