Inverse Laplace Transform of s^2/(s^3 + 6s^2 + 12s + 8):

To find the inverse Laplace transform of the given expression, we can use partial fraction decomposition and the Laplace transform table.

Partial Fraction Decomposition:

We start by factoring the denominator:

s^3 + 6s^2 + 12s + 8 = (s + 2)(s + 2)(s + 2)

Since the denominator has repeated factors, the partial fraction decomposition will have the following form:

s^2/(s^3 + 6s^2 + 12s + 8) = A/(s + 2) + B/(s + 2)^2 + C/(s + 2)^3

Multiplying both sides by the denominator, we get:

s^2 = A(s + 2)(s + 2) + B(s + 2)(s + 2) + C(s + 2)

Expanding and equating coefficients, we have:

s^2 = (A + B)s^2 + (4A + 4B + C)s + 4A + 4B + 2C

Equating coefficients of like powers of s, we get the following system of equations:

A + B = 1 (coefficient of s^2)
4A + 4B + C = 0 (coefficient of s)
4A + 4B + 2C = 0 (constant term)

Solving the system of equations, we find A = 1/4, B = 3/4, and C = -1.

Therefore, the partial fraction decomposition is:

s^2/(s^3 + 6s^2 + 12s + 8) = 1/4/(s + 2) + 3/4/(s + 2)^2 - 1/(s + 2)^3

Inverse Laplace Transform:

Using the Laplace transform table, the inverse Laplace transform of each term can be found:

1/4/(s + 2) transforms to (1/4)e^(-2t)
3/4/(s + 2)^2 transforms to (3/4)te^(-2t)
-1/(s + 2)^3 transforms to -1/2t^2e^(-2t)

Therefore, the inverse Laplace transform of s^2/(s^3 + 6s^2 + 12s + 8) is:

(1/4)e^(-2t) + (3/4)te^(-2t) - (1/2t^2)e^(-2t)

This is the final answer after finding the inverse Laplace transform of the given expression.