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Let's analyze the given functions and the division process.

You have two functions:

- \( s(x) = \frac{x - 4}{x^2 - 25} \)
- \( t(x) = \frac{x - 5}{4 - x} \)

The goal is to find \( \left(\frac{s}{t}\right)(x) \), which involves dividing \( s(x) \) by \( t(x) \).

### Step-by-Step Process:

1. **Function Division**:
   \[
   \left(\frac{s}{t}\right)(x) = \frac{s(x)}{t(x)} = \frac{\frac{x - 4}{x^2 - 25}}{\frac{x - 5}{4 - x}}
   \]
   To simplify this, multiply by the reciprocal of \( t(x) \):
   \[
   \left(\frac{s}{t}\right)(x) = \frac{x - 4}{x^2 - 25} \times \frac{4 - x}{x - 5}
   \]

2. **Simplify the Denominators and Numerators**:
   Factor \( x^2 - 25 \) as \( (x - 5)(x + 5) \):
   \[
   \left(\frac{s}{t}\right)(x) = \frac{x - 4}{(x - 5)(x + 5)} \times \frac{4 - x}{x - 5}
   \]
   Notice that \( 4 - x \) is the negative of \( x - 4 \), so:
   \[
   4 - x = -(x - 4)
   \]
   Substituting:
   \[
   \left(\frac{s}{t}\right)(x) = \frac{x - 4}{(x - 5)(x + 5)} \times \frac{-(x - 4)}{x - 5}
   \]

3. **Simplify the Expression**:
   \[
   \left(\frac{s}{t}\right)(x) = \frac{-(x - 4)^2}{(x - 5)^2(x + 5)}
   \]
   Expanding \( -(x - 4)^2 \), we get:
   \[
   \left(\frac{s}{t}\right)(x) = \frac{-1}{(x + 5)(x - 5)}
   \]

### Domain of the Function:
For the domain, exclude values that make the denominator zero. From the simplified form, the denominator has factors \( (x + 5)(x - 5) \), so the values to exclude are:
   \[
   x = -5 \quad \text{and} \quad x = 5
   \]
Additionally, from \( t(x) = \frac{x - 5}{4 - x} \), exclude \( x = 4 \), as that would make the denominator zero in the original \( t(x) \) function.

### Final Answer:
\[
\boxed{\left(\frac{s}{t}\right)(x) = \frac{-1}{(x + 5)(x - 5)}}
\]
The domain is:
\[
(-\infty, -5) \cup (-5, 4) \cup (4, 5) \cup (5, \infty)
\]

The issue in your attempt might be an algebraic mistake when simplifying the expression or considering the domain. Let me know if you'd like more details on a specific part of the process.

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Here are 5 related questions:

1. How do you find the domain of a rational function in general?
2. What happens when the numerator of a rational function becomes zero?
3. How do you handle factoring polynomials to simplify rational functions?
4. What are some techniques for recognizing and simplifying common factor patterns?
5. How does multiplying by the reciprocal affect the domain in rational function division?

**Tip**: Always check for domain restrictions before and after simplifying a rational expression.

Find the indicated function \( \frac{s}{t} \) for \( s(x) = \frac{x-4}{x^2 - 25} \) and \( t(x) = \frac{x-5}{4-x} \), and determine the domain of the resulting function.

Learn how to divide rational functions and determine their domain. This step-by-step solution simplifies the division of \( s(x) = \frac{x-4}{x^2 - 25} \) by \( t(x) = \frac{x-5}{4-x} \), including factoring and handling domain restrictions. Suitable for high school students.

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