We are tasked with solving for $y(t)$ given $\mathcal{L}[y] = \frac{8s}{(s+7)(s-4)}$. Let's break this down:

Step 1: Decompose the transfer function using partial fraction decomposition

The Laplace transform provided is:

$\mathcal{L}[y] = \frac{8s}{(s+7)(s-4)}.$

We decompose it into partial fractions:

$\frac{8s}{(s+7)(s-4)} = \frac{A}{s+7} + \frac{B}{s-4},$

where $A$ and $B$ are constants to be determined.

Step 2: Solve for $A$ and $B$

Multiply through by the denominator $(s+7)(s-4)$:

$8s = A(s-4) + B(s+7).$

Expanding:

$8s = A s - 4A + B s + 7B.$

Combine like terms:

$8s = (A + B)s + (-4A + 7B).$

Equating coefficients of $s$ and the constant term:

1. $A + B = 8$,
2. $-4A + 7B = 0$.

From the first equation:

$B = 8 - A.$

Substitute $B = 8 - A$ into the second equation:

$-4A + 7(8 - A) = 0.$

Simplify:

$-4A + 56 - 7A = 0,$

$-11A + 56 = 0,$

$A = \frac{56}{11}.$

Using $B = 8 - A$:

$B = 8 - \frac{56}{11} = \frac{88}{11} - \frac{56}{11} = \frac{32}{11}.$

So:

$\frac{8s}{(s+7)(s-4)} = \frac{\frac{56}{11}}{s+7} + \frac{\frac{32}{11}}{s-4}.$

Step 3: Take the inverse Laplace transform

The inverse Laplace transform of each term is:

1. $\mathcal{L}^{-1}\left[\frac{\frac{56}{11}}{s+7}\right] = \frac{56}{11}e^{-7t},$,
2. $\mathcal{L}^{-1}\left[\frac{\frac{32}{11}}{s-4}\right] = \frac{32}{11}e^{4t}.$,

Thus:

$y(t) = \frac{56}{11}e^{-7t} + \frac{32}{11}e^{4t}.$

Final Answer:

$\boxed{y(t) = \frac{56}{11}e^{-7t} + \frac{32}{11}e^{4t}}.$

This corresponds to the first option.

Would you like a detailed breakdown of partial fractions or Laplace transforms? Here are some related questions to explore:

1. What is the significance of partial fraction decomposition in Laplace transforms?
2. How do initial conditions affect Laplace transform solutions?
3. Can you explain the properties of $e^{at}$ in Laplace domain mappings?
4. How does the Laplace transform handle systems of differential equations?
5. Can this method be extended to solve higher-order differential equations?

Tip: Double-check your partial fraction coefficients to ensure the decomposition is accurate before taking the inverse transform.