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The problem set in the image involves expressing each given function as a partial fraction. Below is the solution for each:

### 1. \(\frac{x}{x^2 - x - 6}\)
Factor the denominator:
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]
So, express as:
\[
\frac{x}{x^2 - x - 6} = \frac{A}{x - 3} + \frac{B}{x + 2}
\]

### 2. \(\frac{4x^2 + 4x + 1}{x^2 + 4}\)
Since the degree of the numerator and denominator is the same, long division can be used:
\[
\text{This does not factor into a simple partial fraction form.}
\]

### 3. \(\frac{5 - 3x}{(5 - 3x)(1 + 2x)}\)
This can be simplified directly:
\[
= \frac{1}{1 + 2x}
\]

### 4. \(\frac{x + 3}{x^2 + 3x + 2}\)
Factor the denominator:
\[
x^2 + 3x + 2 = (x + 1)(x + 2)
\]
So, express as:
\[
\frac{x + 3}{(x + 1)(x + 2)} = \frac{A}{x + 1} + \frac{B}{x + 2}
\]

### 5. \(\frac{2}{x^2 - 7x - 15}\)
Factor the denominator:
\[
x^2 - 7x - 15 = (x - 10)(x + 3)
\]
So, express as:
\[
\frac{2}{x^2 - 7x - 15} = \frac{A}{x - 10} + \frac{B}{x + 3}
\]

### 6. \(\frac{x^2}{x^2 - 1}\)
Factor the denominator:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
So, express as:
\[
\frac{x^2}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1}
\]

### 7. \(\frac{2x - 1}{(x + 1)^2}\)
Since the denominator is a repeated linear factor, express as:
\[
\frac{2x - 1}{(x + 1)^2} = \frac{A}{x + 1} + \frac{B}{(x + 1)^2}
\]

### 8. \(\frac{2x^2 + 10x - 3}{(x + 1)(x - 9)^2}\)
Express as:
\[
\frac{2x^2 + 10x - 3}{(x + 1)(x - 9)^2} = \frac{A}{x + 1} + \frac{B}{x - 9} + \frac{C}{(x - 9)^2}
\]

### 9. \(\frac{5x^2 + 23x - 7}{(x + 1)(x + 3)^2}\)
Express as:
\[
\frac{5x^2 + 23x - 7}{(x + 1)(x + 3)^2} = \frac{A}{x + 1} + \frac{B}{x + 3} + \frac{C}{(x + 3)^2}
\]

### 10. \(\frac{11x^2 + 4x + 12}{(x + 1)(x + 4)}\)
Express as:
\[
\frac{11x^2 + 4x + 12}{(x + 1)(x + 4)} = \frac{A}{x + 1} + \frac{B}{x + 4}
\]

### 11. \(\frac{3x^2 + 5x + 1}{(x - 2)(3x - 2)}\)
Express as:
\[
\frac{3x^2 + 5x + 1}{(x - 2)(3x - 2)} = \frac{A}{x - 2} + \frac{B}{3x - 2}
\]

### 12. \(\frac{x^3 - 1}{(x + 1)(x - 2)}\)
Factor \(x^3 - 1\):
\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]
Express as:
\[
\frac{x^3 - 1}{(x + 1)(x - 2)} = \frac{A(x^2 + x + 1)}{(x + 1)(x - 2)}
\]

Would you like detailed solutions for any specific question or more clarification? 

### Related Questions
1. How do you determine the constants \(A\) and \(B\) in partial fractions?
2. Can you factor higher-degree polynomials for partial fractions?
3. What are some examples where partial fraction decomposition is used in real-life problems?
4. How do you solve partial fractions with repeated linear factors?
5. What is the significance of partial fractions in calculus, particularly in integration?

### Tip
Always double-check your factorization before beginning the partial fraction decomposition to avoid errors in the setup.

State each of the following fraction functions as a partial fraction.

Learn how to express algebraic fractions as partial fractions, involving polynomial factorization and application of the Fundamental Theorem of Algebra. This detailed step-by-step guide covers multiple examples suitable for students in Grades 10-12.

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