The image you've uploaded contains a series of problems related to Laplace transforms (denoted by the operator $\mathcal{L}$). I'll walk through solving the first problem to demonstrate the approach.

Problem 1:

Find the Laplace transform of $2 + e^{-3t} + \sin(4t)$.

Solution:

Using standard Laplace transform rules:

1. For the constant term $2$: $\mathcal{L}(2) = \frac{2}{s}$

2. For the exponential term $e^{-3t}$: $\mathcal{L}(e^{-3t}) = \frac{1}{s+3}$

3. For the sine function $\sin(4t)$: $\mathcal{L}(\sin(4t)) = \frac{4}{s^2 + 16}$

Now combine the results: $\mathcal{L}(2 + e^{-3t} + \sin(4t)) = \frac{2}{s} + \frac{1}{s+3} + \frac{4}{s^2 + 16}$

Looking at the answer choices, the correct one is (d).

Would you like me to continue solving the other problems or clarify any details? Here are five related questions that expand on this concept:

1. What is the Laplace transform of $\cos(5t)$?
2. How does the shifting theorem apply to $e^{-3t} \cdot \sin(4t)$?
3. Can you find the Laplace transform of a unit step function $u(t - a) \cdot f(t - a)$?
4. What is the inverse Laplace transform of $\frac{1}{s^2 + a^2}$?
5. How does the Laplace transform of a derivative work?

Tip: Remember that shifting in the time domain (like $e^{-at}$) corresponds to a shift in the $s$-domain for Laplace transforms.