The image contains a set of Laplace transform-related multiple-choice problems. Let's go through the questions step by step, starting with the first one:

1. $\mathcal{L}\{2 + e^{-3t} + \sin 4t\}$

The Laplace transform of:

• $2$ is $\frac{2}{s}$,
• $e^{-3t}$ is $\frac{1}{s + 3}$,
• $\sin(4t)$ is $\frac{4}{s^2 + 16}$.

Summing them up, we get: $\frac{2}{s} + \frac{1}{s + 3} + \frac{4}{s^2 + 16}$ The correct answer is (d).

2. $\mathcal{L}\{\sin(3t - 2)\}$

Using the phase shift property of the Laplace transform: $\mathcal{L}\{\sin(3t)\} = \frac{3}{s^2 + 9}$ But due to the time-shift, we apply Euler's identity, which leads to additional terms. The correct answer appears to be (d).

3. $\mathcal{L}\{t \sin(2t)\}$

Using the formula $\mathcal{L}\{t \sin(at)\} = \frac{2s}{(s^2 + a^2)^2}$ for $a = 2$: The correct answer is (b).

4. $\mathcal{L}\{e^{3t} \cosh(2t)\}$

We use the Laplace transform of $\cosh(at)$, adjusted by the exponential term: The correct answer is (b).

5. $\mathcal{L}\{t \sin(4t)\}$

From the known formula for $t \sin(at)$, the Laplace transform is: $\mathcal{L}\{t \sin(at)\} = \frac{2s}{(s^2 + a^2)^2}$ For $a = 4$, the correct answer is (a).

Do you want me to continue with more detailed steps, or do you have any questions about the explanation so far?

Here are 5 related questions:

1. What is the Laplace transform of $\cos(at)$?
2. How do phase shifts affect the Laplace transform?
3. What is the inverse Laplace transform of $\frac{1}{s^2 + a^2}$?
4. How does the time-shift theorem modify the Laplace transform?
5. What is the Laplace transform of $t^n e^{at}$?

Tip: Use known Laplace transform tables for common functions to speed up solving problems!