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Let's solve this problem step by step to find the derivative of \( f(x) = 2x^2 + 9x \) using the four-step process outlined in the image.

### Step 1: 
We need to find \( f(x + h) \).

\[
f(x + h) = 2(x + h)^2 + 9(x + h)
\]
Now expand the expression:
\[
= 2(x^2 + 2xh + h^2) + 9(x + h)
\]
\[
= 2x^2 + 4xh + 2h^2 + 9x + 9h
\]

So, 
\[
f(x + h) = 2x^2 + 4xh + 2h^2 + 9x + 9h
\]

### Step 2:
We need to find \( f(x + h) - f(x) \). Recall that \( f(x) = 2x^2 + 9x \).

\[
f(x + h) - f(x) = (2x^2 + 4xh + 2h^2 + 9x + 9h) - (2x^2 + 9x)
\]
Simplifying:
\[
f(x + h) - f(x) = 4xh + 2h^2 + 9h
\]

### Step 3:
Now we need to find \( \frac{f(x + h) - f(x)}{h} \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{4xh + 2h^2 + 9h}{h}
\]
Simplifying by dividing each term by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = 4x + 2h + 9
\]

### Step 4:
Finally, we take the limit as \( h \to 0 \) to find \( f'(x) \):

\[
f'(x) = \lim_{h \to 0} (4x + 2h + 9)
\]
Since the term \( 2h \) goes to 0 as \( h \to 0 \), we are left with:
\[
f'(x) = 4x + 9
\]

### Final Answer:
The derivative of \( f(x) = 2x^2 + 9x \) is:
\[
f'(x) = 4x + 9
\]

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Do you need any further clarification on any step, or would you like additional details?

Here are 5 related questions to help deepen your understanding:
1. How does the four-step process relate to the definition of a derivative?
2. How would you find the tangent line equation using \( f'(x) \)?
3. What is the geometric interpretation of the derivative \( f'(x) \)?
4. How would the result change if the function had a constant term?
5. Can you apply the same steps to find the derivative of \( f(x) = 3x^3 + 2x \)?

**Tip:** When finding derivatives using the four-step process, always simplify early to avoid dealing with overly complex expressions later.

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. f(x) = 2x^2 + 9x.

Learn how to find the derivative of f(x) = 2x^2 + 9x using the four-step process. This solution involves expanding terms, simplifying expressions, and applying the limit definition of a derivative. Suitable for students in Grades 11-12 studying calculus.

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