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The problem asks to find the Laplace Transform \( L \left[ \left( \sqrt{t} - 1 \right)^2 \right] \).

### Step-by-Step Solution:

1. **Expand the Expression:**
   \[
   \left( \sqrt{t} - 1 \right)^2 = t - 2\sqrt{t} + 1
   \]
   
2. **Take the Laplace Transform of Each Term Separately:**
   - The Laplace Transform of \( t \) is:
     \[
     L[t] = \frac{1}{s^2}
     \]
   - The Laplace Transform of \( \sqrt{t} \) is:
     \[
     L[\sqrt{t}] = \frac{\sqrt{\pi}}{2s^{3/2}}
     \]
   - The Laplace Transform of \( 1 \) is:
     \[
     L[1] = \frac{1}{s}
     \]
   
3. **Put It All Together:**
   \[
   L \left[ t - 2\sqrt{t} + 1 \right] = \frac{1}{s^2} - 2 \cdot \frac{\sqrt{\pi}}{2s^{3/2}} + \frac{1}{s}
   \]
   Simplify:
   \[
   L \left[ t - 2\sqrt{t} + 1 \right] = \frac{1}{s^2} - \frac{\sqrt{\pi}}{s^{3/2}} + \frac{1}{s}
   \]

### Final Answer:
\[
L \left[ \left( \sqrt{t} - 1 \right)^2 \right] = \frac{1}{s^2} - \frac{\sqrt{\pi}}{s^{3/2}} + \frac{1}{s}
\]

Would you like more details or have any questions?

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Here are 5 related questions you can explore:
1. What is the Laplace transform of \( t^n \) for any integer \( n \)?
2. How do you compute the Laplace transform of fractional powers of \( t \)?
3. What is the significance of Laplace transforms in differential equations?
4. Can we find the inverse Laplace transform of \( \frac{1}{s^2} \)?
5. How does the Laplace transform handle exponential decay functions?

**Tip:** The Laplace transform is especially useful for converting differential equations into algebraic equations, simplifying their solution.

Step-by-step guide to finding the Laplace Transform of the function (√t - 1)^2. This problem involves algebraic expansion and applying the Laplace Transform to each term separately. Suitable for undergraduate students in math or engineering.

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