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The problem asks to find and simplify the difference quotient \(\frac{f(x+h) - f(x)}{h}\) for the given function \(f(x) = x^2 - 9x - 2\).

Let's break it down:

### Step 1: Find \(f(x + h)\)
We need to substitute \(x + h\) into the function \(f(x)\).

\[
f(x + h) = (x + h)^2 - 9(x + h) - 2
\]

Expanding the terms:
\[
f(x + h) = (x^2 + 2xh + h^2) - 9(x + h) - 2
\]
\[
f(x + h) = x^2 + 2xh + h^2 - 9x - 9h - 2
\]

### Step 2: Compute the difference \(f(x + h) - f(x)\)
Now, subtract \(f(x)\) from \(f(x + h)\):

\[
f(x + h) - f(x) = \left(x^2 + 2xh + h^2 - 9x - 9h - 2\right) - \left(x^2 - 9x - 2\right)
\]
Distribute the negative sign and simplify:
\[
f(x + h) - f(x) = x^2 + 2xh + h^2 - 9x - 9h - 2 - x^2 + 9x + 2
\]
Cancel out the common terms \(x^2\), \(-9x\), and \(-2\):
\[
f(x + h) - f(x) = 2xh + h^2 - 9h
\]

### Step 3: Simplify the difference quotient
Now, divide by \(h\):

\[
\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2 - 9h}{h}
\]

Factor out \(h\) from the numerator:
\[
\frac{f(x + h) - f(x)}{h} = \frac{h(2x + h - 9)}{h}
\]

Cancel \(h\) from the numerator and denominator (assuming \(h \neq 0\)):
\[
\frac{f(x + h) - f(x)}{h} = 2x + h - 9
\]

### Answer:
The simplified form of the difference quotient is:
\[
2x + h - 9
\]

This matches the last option in the multiple-choice list, so the correct answer is **2x + h - 9**.

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as \(h\) approaches 0?
3. Can you compute the derivative of \(f(x)\) directly using the power rule?
4. What is the geometric interpretation of the difference quotient?
5. How does this problem change if the function was cubic instead of quadratic?

**Tip:** The difference quotient is foundational in calculus, leading to the concept of the derivative, which gives the slope of the tangent line to the curve at any point.

Given the function f(x) = x^2 - 9x - 2, find and simplify the difference quotient f(x+h) - f(x) / h.

Simplify the difference quotient for the quadratic function f(x) = x^2 - 9x - 2. This step-by-step solution covers key concepts in calculus such as the difference quotient and limits, suitable for high school students in Grades 10-12.

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