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The question asks to use the four-step process to find the derivative \( f'(x) \) of the function \( f(x) = x^2 + 3x - 2 \) and then to find \( f'(1) \), \( f'(2) \), and \( f'(3) \).

### The four-step process for finding the derivative is:

1. **Step 1**: Find \( f(x + h) \), where \( f(x) = x^2 + 3x - 2 \).
   
2. **Step 2**: Compute the difference \( f(x + h) - f(x) \).

3. **Step 3**: Divide the difference by \( h \), i.e., find \( \frac{f(x + h) - f(x)}{h} \).

4. **Step 4**: Take the limit as \( h \to 0 \) of the expression from Step 3 to get \( f'(x) \).

Let's apply this step by step.

### Step 1: Find \( f(x + h) \)
\[
f(x + h) = (x + h)^2 + 3(x + h) - 2
\]
Expanding the terms:
\[
f(x + h) = (x^2 + 2xh + h^2) + (3x + 3h) - 2
\]
\[
f(x + h) = x^2 + 2xh + h^2 + 3x + 3h - 2
\]

### Step 2: Compute \( f(x + h) - f(x) \)
Now, subtract \( f(x) = x^2 + 3x - 2 \) from \( f(x + h) \):
\[
f(x + h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h - 2) - (x^2 + 3x - 2)
\]
Simplifying:
\[
f(x + h) - f(x) = 2xh + h^2 + 3h
\]

### Step 3: Divide by \( h \)
Now, divide the expression by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 + 3h}{h}
\]
Simplifying:
\[
\frac{f(x + h) - f(x)}{h} = 2x + h + 3
\]

### Step 4: Take the limit as \( h \to 0 \)
Now, take the limit as \( h \to 0 \):
\[
f'(x) = \lim_{h \to 0} (2x + h + 3) = 2x + 3
\]

Thus, the derivative \( f'(x) = 2x + 3 \).

### Now, find \( f'(1) \), \( f'(2) \), and \( f'(3) \):

- \( f'(1) = 2(1) + 3 = 2 + 3 = 5 \)
- \( f'(2) = 2(2) + 3 = 4 + 3 = 7 \)
- \( f'(3) = 2(3) + 3 = 6 + 3 = 9 \)

### Final Answers:
- \( f'(1) = 5 \)
- \( f'(2) = 7 \)
- \( f'(3) = 9 \)

Would you like more details on any of the steps or have any further questions?

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Here are five related questions to explore further:
1. How would the four-step process change for a function with a higher degree, like \( f(x) = x^3 + 2x^2 - x \)?
2. How is the four-step process related to the definition of a derivative in calculus?
3. What would the second derivative \( f''(x) \) be for this function?
4. How would the derivative change if the constant term was removed from the function?
5. What happens if you evaluate the derivative at negative values of \( x \), like \( f'(-1) \)?

**Tip**: The four-step process is a formal way to derive the derivative using the definition of limits, but you can often take shortcuts using known derivative rules.

Use the four-step process to find f'(x) and then find f'(1), f'(2), and f'(3) for the function f(x) = x^2 + 3x - 2.

This page explains how to find the derivative of the quadratic function f(x) = x^2 + 3x - 2 using the four-step process. It includes a detailed step-by-step solution, culminating in the evaluations of f'(1), f'(2), and f'(3). This problem is ideal for high school calculus students in Grades 10-12.

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