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Let's work on each problem step-by-step.

### Problem 1

The expression is:
\[
\frac{5x + 1}{x^3 - x}
\]

1. **Factor the denominator:** We can factor \( x^3 - x \) as follows:
   \[
   x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1)
   \]

2. **Set up the partial fraction decomposition:** Since the factors are \( x \), \( x - 1 \), and \( x + 1 \), we can write:
   \[
   \frac{5x + 1}{x(x - 1)(x + 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1}
   \]

3. **Clear the denominators:** Multiply both sides by \( x(x - 1)(x + 1) \) to obtain:
   \[
   5x + 1 = A(x - 1)(x + 1) + B(x)(x + 1) + C(x)(x - 1)
   \]

4. **Expand and solve for \( A \), \( B \), and \( C \)** by comparing coefficients.

### Problem 2

The expression is:
\[
\frac{x^3 - x^2 + 2x + 4}{x^2 (x - 2)^2}
\]

1. **Factor the denominator:** The denominator is already factored as \( x^2 (x - 2)^2 \).

2. **Set up the partial fraction decomposition:** Since we have a repeated factor \( x \) and \( (x - 2) \), we write:
   \[
   \frac{x^3 - x^2 + 2x + 4}{x^2 (x - 2)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 2} + \frac{D}{(x - 2)^2}
   \]

3. **Clear the denominators:** Multiply both sides by \( x^2 (x - 2)^2 \) and then expand and compare coefficients to solve for \( A \), \( B \), \( C \), and \( D \).

Would you like detailed solutions for each of these steps, or do you have any questions?

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### Additional Questions for Deeper Understanding

1. How do we determine the form of partial fractions when we have repeated linear factors?
2. Why do we use different constants (like \( A \), \( B \), etc.) for each term in the decomposition?
3. What happens if a polynomial in the numerator has a higher degree than the denominator?
4. How can we check if our partial fraction decomposition is correct?
5. How does partial fraction decomposition apply in integral calculus?

### Tip
When decomposing fractions, always start by factoring the denominator completely—this simplifies the setup of partial fractions.

Decompose (5x + 1) / (x^3 - x) into a sum of partial fractions.
Decompose (x^3 - x^2 + 2x + 4) / (x^2 * (x - 2)^2) into a sum of partial fractions.

Learn how to decompose rational expressions like (5x + 1) / (x^3 - x) and (x^3 - x^2 + 2x + 4) / (x^2 * (x - 2)^2) into partial fractions. This step-by-step solution covers algebraic techniques for handling distinct and repeated linear factors, suitable for students in Grades 10-12.

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