We are asked to determine the correct form of the partial fraction decomposition for the expression:

$\frac{x^2 + 9}{(x - 1)^2(x^2 + 2x + 5)(x^2 - x - 6)}$

Denominator Factorization:

1. $(x - 1)^2$ is a repeated linear factor.
2. $x^2 + 2x + 5$ is an irreducible quadratic factor.
3. $x^2 - x - 6 = (x - 3)(x + 2)$ is factored as two distinct linear factors.

Decomposition Structure:

The partial fraction decomposition must include:

• Terms for the linear factor $x - 1$, both in simple and repeated form: $\frac{A}{x - 1} + \frac{B}{(x - 1)^2}$.
• A term for the irreducible quadratic $x^2 + 2x + 5$: $\frac{Bx + C}{x^2 + 2x + 5}$.
• Terms for the distinct linear factors $x - 3$ and $x + 2$: $\frac{D}{x - 3}$ and $\frac{E}{x + 2}$.

Answer Choice:

Based on this structure, option B is correct. It contains: $\frac{A}{x - 1} + \frac{B}{(x - 1)^2} + \frac{Bx + C}{x^2 + 2x + 5} + \frac{D}{x - 3} + \frac{E}{x + 2}$

Would you like more details on how partial fraction decomposition works, or do you have any other questions?

Here are 5 related questions to expand on this:

1. How do you perform partial fraction decomposition for repeated quadratic factors?
2. What are the steps to solve for constants $A, B, C, D,$ and $E$ after decomposition?
3. Can you explain how to deal with irreducible quadratics in decompositions?
4. What happens when the denominator has higher powers of irreducible quadratics?
5. How does partial fraction decomposition help in solving integrals?

Tip: Always check if the denominator is factored completely before setting up the partial fractions.