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The problem asks to find the Laplace transform of the expression:

\[
(1 + 2t - 3t^2 + 4t^3)H(t - 2)
\]

where \( H(t - 2) \) is the Heaviside step function shifted by 2.

### Steps to solve:

1. **Break down the expression**: The Laplace transform of a function multiplied by \( H(t - a) \) results in a time shift by \( a \). Specifically, the formula for this is:

   \[
   \mathcal{L}\{ f(t - a) H(t - a) \} = e^{-as} \mathcal{L}\{ f(t) \}
   \]

   Here, the Heaviside function \( H(t - 2) \) causes a shift by 2 units. So, we can rewrite the expression as:

   \[
   \mathcal{L}\{(1 + 2t - 3t^2 + 4t^3) H(t - 2)\} = e^{-2s} \mathcal{L}\{ 1 + 2(t + 2) - 3(t + 2)^2 + 4(t + 2)^3 \}
   \]

2. **Apply the Laplace Transform to the polynomial terms**:
   We first calculate the Laplace transform for each term of \( 1 + 2t - 3t^2 + 4t^3 \). The general Laplace transform of \( t^n \) is given by:

   \[
   \mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}
   \]

   So:
   - \( \mathcal{L}\{1\} = \frac{1}{s} \)
   - \( \mathcal{L}\{2t\} = \frac{2}{s^2} \)
   - \( \mathcal{L}\{-3t^2\} = -\frac{6}{s^3} \)
   - \( \mathcal{L}\{4t^3\} = \frac{24}{s^4} \)

3. **Combine the results**:
   The Laplace transform of \( 1 + 2t - 3t^2 + 4t^3 \) is:

   \[
   \mathcal{L}\{ 1 + 2t - 3t^2 + 4t^3 \} = \frac{1}{s} + \frac{2}{s^2} - \frac{6}{s^3} + \frac{24}{s^4}
   \]

4. **Include the Heaviside function**:
   Multiplying by \( e^{-2s} \) for the time shift:

   \[
   \mathcal{L}\{(1 + 2t - 3t^2 + 4t^3) H(t - 2)\} = e^{-2s} \left( \frac{1}{s} + \frac{2}{s^2} - \frac{6}{s^3} + \frac{24}{s^4} \right)
   \]

Thus, the final Laplace transform is:

\[
e^{-2s} \left( \frac{1}{s} + \frac{2}{s^2} - \frac{6}{s^3} + \frac{24}{s^4} \right)
\]

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Do you want more details or have any questions? Here are some relative questions to expand on this topic:

1. What is the significance of the Heaviside step function in Laplace transforms?
2. How does the Laplace transform handle time shifts in general?
3. What is the Laplace transform of more complex functions involving exponentials or trigonometric terms?
4. How is the inverse Laplace transform computed for such expressions?
5. Can the Laplace transform be used to solve differential equations involving step functions?

**Tip**: When solving Laplace transform problems involving the Heaviside function, always apply the time-shifting property before computing the transform of the main function.

Find the Laplace transform of (1 + 2t - 3t^2 + 4t^3)H(t - 2).

Learn how to find the Laplace transform of the expression (1 + 2t - 3t^2 + 4t^3)H(t - 2). This step-by-step solution involves applying the time-shifting property of the Laplace transform to a polynomial function and the Heaviside step function. Suitable for university-level students studying differential equations and control systems.

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