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To solve for the Laplace transform of the given function \( f(t) = 6 + u(t - 3) \cdot (t + 9) \), let's break it down:

### Step 1: Laplace Transform of Constant Term
The first term is a constant:
\[
\mathcal{L}\{6\} = \frac{6}{s}
\]
since the Laplace transform of a constant \( c \) is \( \frac{c}{s} \).

### Step 2: Laplace Transform of Heaviside Function
The second part involves a Heaviside step function \( u(t - 3) \) multiplied by \( (t + 9) \). We need to account for the step occurring at \( t = 3 \), which implies a shift in the time domain.

This can be handled using the **time-shifting property** of Laplace transforms:
\[
\mathcal{L}\{ u(t - a) \cdot f(t - a) \} = e^{-as} \cdot \mathcal{L}\{f(t)\}
\]
For the term \( u(t - 3) \cdot (t + 9) \), let's rewrite \( (t + 9) \) as \( (t - 3) + 12 \). So, we can break this into two parts:
\[
u(t - 3) \cdot (t + 9) = u(t - 3) \cdot \left[(t - 3) + 12\right]
\]
This means we have two sub-terms to deal with:
1. \( u(t - 3) \cdot (t - 3) \)
2. \( u(t - 3) \cdot 12 \)

#### Sub-term 1: \( u(t - 3) \cdot (t - 3) \)
This becomes the Laplace transform of \( t \), but shifted by 3:
\[
\mathcal{L}\{u(t - 3) \cdot (t - 3)\} = e^{-3s} \cdot \frac{1}{s^2}
\]

#### Sub-term 2: \( u(t - 3) \cdot 12 \)
This becomes a constant with a shift:
\[
\mathcal{L}\{u(t - 3) \cdot 12\} = e^{-3s} \cdot \frac{12}{s}
\]

### Step 3: Combine Everything
Now, putting all the pieces together:
\[
F(s) = \frac{6}{s} + e^{-3s} \cdot \left( \frac{1}{s^2} + \frac{12}{s} \right)
\]

This is the Laplace transform of the given function.

### Final Answer:
\[
F(s) = \frac{6}{s} + e^{-3s} \cdot \left( \frac{1}{s^2} + \frac{12}{s} \right)
\]

Would you like a more detailed explanation of any of these steps?

Here are 5 related questions to deepen your understanding:
1. How does the time-shifting property of the Laplace transform work in general?
2. What is the Laplace transform of \( t^n \) and how is it derived?
3. How do you use the convolution theorem in Laplace transforms?
4. How can you use Laplace transforms to solve differential equations?
5. What are the key differences between using the Heaviside function and Dirac delta in Laplace transforms?

**Tip:** Always watch for functions involving the Heaviside step function, as they often signal the need for the time-shifting property in Laplace transforms.

Find the Laplace transform of f(t) = 6 + u(t - 3) * (t + 9).

Learn how to compute the Laplace transform of f(t) = 6 + u(t - 3) * (t + 9), using key concepts like the time-shifting property and basic Laplace transforms. Suitable for undergraduate students.

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