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We are given the expression \( L^{-1} \left\{ \frac{1}{(s^2 + 16)(s^2 + 25)} \right\} \) and asked to select the correct inverse Laplace transformation from the multiple choices.

### Steps for finding the inverse Laplace:
1. **Factor the denominator:** 
   - \( s^2 + 16 \) is related to the Laplace transform of \( \sin(4t) \), because \( s^2 + a^2 \) corresponds to \( \frac{\sin(at)}{a} \).
   - \( s^2 + 25 \) is related to the Laplace transform of \( \sin(5t) \), because \( s^2 + a^2 \) corresponds to \( \frac{\sin(at)}{a} \).

2. **Partial Fraction Decomposition:**
   Break down the expression into partial fractions of the form:
   \[
   \frac{1}{(s^2 + 16)(s^2 + 25)} = \frac{A}{s^2 + 16} + \frac{B}{s^2 + 25}
   \]
   Solve for \( A \) and \( B \).

3. **Inverse Laplace:**
   Once the partial fractions are determined, use the known inverse Laplace transforms:
   \[
   \mathcal{L}^{-1} \left\{ \frac{1}{s^2 + a^2} \right\} = \frac{\sin(at)}{a}
   \]

### Apply the process:
From the denominator, we identify that \( s^2 + 16 \) gives \( \sin(4t) \), and \( s^2 + 25 \) gives \( \sin(5t) \). So, the terms involving \( \sin(4t) \) and \( \sin(5t) \) are correct. After performing partial fractions and applying inverse Laplace:

- The correct expression is a combination of \( \frac{\sin(4t)}{4} \) and \( \frac{\sin(5t)}{5} \).

### The correct answer is **(b)**:
\[
\frac{\sin(5t)}{5} - \frac{\sin(4t)}{4}
\]

Would you like me to break down the partial fraction decomposition in more detail?

Here are some follow-up questions for deeper understanding:
1. How is the partial fraction decomposition applied to rational Laplace transforms?
2. What is the general formula for the inverse Laplace transform of \( \frac{1}{s^2 + a^2} \)?
3. Why is the inverse Laplace transform of \( \frac{1}{s^2 + a^2} \) related to sine functions?
4. How can you solve for constants in partial fraction decomposition?
5. What are other common forms of inverse Laplace transforms involving sine or cosine functions?

**Tip:** When handling rational functions in Laplace transforms, partial fractions help break the expression into simpler terms, making it easier to apply known inverse Laplace transforms.

Learn how to solve the inverse Laplace transform \( L^{-1} \left\{ \frac{1}{(s^2 + 16)(s^2 + 25)} \right\} \) using partial fraction decomposition. This detailed solution includes the key formulas for inverse Laplace transforms involving sine functions. Suitable for university-level students.

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