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.