To find the LCD (Least Common Denominator) of the given rational expressions, we need to:

1. Factor the denominators of each fraction.
2. Identify the unique factors and their highest powers.
3. Multiply these factors together to find the LCD.

Step 1: Factor each denominator

1. For $\frac{x}{x^2 - 16}$: $x^2 - 16 = (x - 4)(x + 4)$ The factored denominator is $(x - 4)(x + 4)$.

2. For $\frac{4}{x^2}$: The denominator $x^2$ is already factored.

3. For $\frac{4}{20 - 5x}$: Factor $20 - 5x$: $20 - 5x = -5(x - 4)$ The factored denominator is $-5(x - 4)$.

Step 2: Combine unique factors

The unique factors in the denominators are:

• $(x - 4)$ (appears in $(x - 4)(x + 4)$ and $-5(x - 4)$)
• $(x + 4)$ (from $(x - 4)(x + 4)$)
• $x$ (from $x^2$)
• Constant $5$ (from $-5$ in $-5(x - 4)$) — included as part of scaling if needed.

The highest powers of these factors:

• $x^2$ (from $x^2$),
• $(x - 4)$ (from $(x - 4)(x + 4)$),
• $(x + 4)$ (from $(x - 4)(x + 4)$).

Step 3: Write the LCD in factored form

The LCD is: $\text{LCD} = x^2 (x - 4)(x + 4)$

This is the least common denominator of the given rational expressions.

Would you like a detailed explanation of how this LCD applies to each term? Here are 5 related questions for practice:

1. How do you use the LCD to combine the given rational expressions into a single fraction?
2. What happens if one of the denominators includes an additional factor, e.g., $(x - 4)^2$?
3. How can you verify the correctness of an LCD in complex rational expressions?
4. Why do we include the highest powers of each unique factor when finding the LCD?
5. How does factoring negative constants, like $-5(x - 4)$, affect the LCD?

Tip: Always double-check your factorizations, as a small mistake in factoring can lead to an incorrect LCD.