The problem seems to involve analysis of a function $f(x) = -x - \frac{4}{x}$. It asks about limits of the function, properties related to the derivative $f'(x)$, and related analysis. The image is difficult to read, but I will attempt to answer the questions based on the discernible parts. The available choices are labeled with (C) and (D).

1. Problem Description

The problem seems to involve analysis of a function $f(x) = -x - \frac{4}{x}$. It asks about limits of the function, properties related to the derivative $f'(x)$, and related analysis. The image is difficult to read, but I will attempt to answer the questions based on the discernible parts. The available choices are labeled with (C) and (D).

2. Solution Steps

Without clear, translated questions, I will solve for the first derivative of $f(x)$ and analyze its properties.

$f(x) = -x - \frac{4}{x} = -x - 4x^{-1}$

$f'(x) = -1 - 4(-1)x^{-2} = -1 + 4x^{-2} = -1 + \frac{4}{x^2} = \frac{4 - x^2}{x^2}$

To find where $f'(x) < 0$, we analyze $\frac{4-x^2}{x^2} < 0$. Since $x^2 > 0$ for all $x \neq 0$, we only need to consider $4 - x^2 < 0$, which means $x^2 > 4$. Thus, $x > 2$ or $x < -2$.

The condition $x^2 + 4\ln(x) > 0$ combined with $x > 0$ suggests domain analysis.

3. Final Answer

Due to the image quality and the unreadable language I can't confidently answer the questions related to choosing between the options (C) and (D) or (L) and (H). However, I found $f'(x) = \frac{4-x^2}{x^2}$ and $f'(x) < 0$ when $x > 2$ or $x < -2$.