Laplace Transform and its Definition:

The Laplace transform is a mathematical operation that transforms a function of a real variable t into a function of a complex variable s. It is commonly used in engineering and physics to solve differential equations and analyze systems in the frequency domain.

The Laplace transform of a function f(t), denoted as F(s), is defined as:

F(s) = L{f(t)} = ∫[0 to ∞] e^(-st) f(t) dt

where s is a complex variable and t is the time variable.

Given Function and its Laplace Transform:

In this problem, we are given the function f(t) = 2t^2e^(-t) and we need to find its Laplace transform F(s).

To find the Laplace transform, we substitute the given function into the Laplace transform definition:

F(s) = L{f(t)} = ∫[0 to ∞] e^(-st) (2t^2e^(-t)) dt

Simplifying the expression inside the integral:

F(s) = 2 ∫[0 to ∞] t^2 e^(-st) e^(-t) dt
= 2 ∫[0 to ∞] t^2 e^(-(s+1)t) dt

Integration by Parts:

To solve this integral, we can use the technique of integration by parts. The formula for integration by parts is:

∫u dv = uv - ∫v du

We choose u = t^2 and dv = e^(-(s+1)t) dt. Then, we can find du and v by differentiating and integrating u and dv, respectively.

du = 2t dt
v = -1/(s+1) e^(-(s+1)t)

Substituting these values into the integration by parts formula:

∫ t^2 e^(-(s+1)t) dt = (-t^2/(s+1)) e^(-(s+1)t) - ∫ (-2t/(s+1)) e^(-(s+1)t) dt

Recursive Integration by Parts:

We can see that we have a similar integral on the right-hand side of the equation. We can apply integration by parts again to solve it.

Let's denote the second integral as I. We choose u = -2t/(s+1) and dv = e^(-(s+1)t) dt. Then, we can find du and v by differentiating and integrating u and dv, respectively.

du = -2/(s+1) dt
v = -1/(s+1) e^(-(s+1)t)

Substituting these values into the integration by parts formula:

I = (-2t/(s+1)) e^(-(s+1)t) - ∫ (-2/(s+1)) (-1/(s+1)) e^(-(s+1)t) dt
= (-2t/(s+1)) e^(-(s+1)t) + (2/(s+1))^2 e^(-(s+1)t) - ∫ (2/(s+1))^2 e^(-(s+1)t) dt

We can observe that we have another similar integral on the right-hand side.