The problem involves determining which of the given expressions has a domain of all real numbers. Let’s analyze each option step by step.

The options and their domains:

1. Option (a): $\frac{1}{a(a + 4)}$

   • This expression is undefined when $a = 0$ (since division by zero is not allowed).
   • It is also undefined when $a + 4 = 0$, or $a = -4$.
   • Domain: All real numbers except $a = 0$ and $a = -4$.

2. Option (b): $\frac{7}{|a| - 4}$

   • This expression is undefined when $|a| - 4 = 0$, or $|a| = 4$.
   • This occurs when $a = 4$ or $a = -4$.
   • Domain: All real numbers except $a = 4$ and $a = -4$.

3. Option (c): $\frac{5}{a - 4}$

   • This expression is undefined when $a - 4 = 0$, or $a = 4$.
   • Domain: All real numbers except $a = 4$.

4. Option (d): $\frac{a}{a + 4}$

   • This expression is undefined when $a + 4 = 0$, or $a = -4$.
   • Domain: All real numbers except $a = -4$.

Correct Answer:

The correct choice is the one whose domain includes all real numbers.
None of the options have a domain of all real numbers, as they all contain restrictions due to division by zero.

Do you have questions about this? Would you like a detailed breakdown of any specific option?

Here are 5 related questions for further exploration:

1. How do you find the domain of more complex rational expressions?
2. What happens to the domain if the denominator includes an absolute value?
3. How can we solve for restrictions when multiple terms are multiplied in the denominator?
4. How does domain restriction change with square root terms?
5. Why does division by zero create undefined points in mathematics?

Tip: Always analyze both the numerator and denominator when finding the domain of rational expressions!