We are tasked with finding and simplifying the difference quotient for the function:

$f(x) = -9x^2 + 2x + 8$

The difference quotient is given by the expression:

$\frac{f(x + h) - f(x)}{h} \quad \text{for} \quad h \neq 0$

Step 1: Find $f(x + h)$

We start by substituting $x + h$ into the function $f(x)$:

$f(x + h) = -9(x + h)^2 + 2(x + h) + 8$

Now, let's expand and simplify:

$f(x + h) = -9[(x + h)^2] + 2(x + h) + 8$

First, expand $(x + h)^2$:

$(x + h)^2 = x^2 + 2xh + h^2$

So,

$f(x + h) = -9(x^2 + 2xh + h^2) + 2(x + h) + 8$ $= -9x^2 - 18xh - 9h^2 + 2x + 2h + 8$

Step 2: Find $f(x + h) - f(x)$

Now subtract $f(x)$ from $f(x + h)$:

$f(x + h) - f(x) = (-9x^2 - 18xh - 9h^2 + 2x + 2h + 8) - (-9x^2 + 2x + 8)$

Distribute the negative sign:

$f(x + h) - f(x) = -9x^2 - 18xh - 9h^2 + 2x + 2h + 8 + 9x^2 - 2x - 8$

Now, combine like terms:

$f(x + h) - f(x) = (-18xh - 9h^2 + 2h)$

Step 3: Simplify the difference quotient

Now, divide $f(x + h) - f(x)$ by $h$:

$\frac{f(x + h) - f(x)}{h} = \frac{-18xh - 9h^2 + 2h}{h}$

Factor out $h$ from the numerator:

$\frac{-18xh - 9h^2 + 2h}{h} = -18x - 9h + 2$

Thus, the simplified difference quotient is:

$\boxed{-18x - 9h + 2}$

This is the simplified form of the difference quotient for the given function.

Would you like any further details or have any questions?

Here are some related questions for further exploration:

1. What is the purpose of the difference quotient in calculus?
2. How does the difference quotient relate to the derivative of a function?
3. Can you apply the difference quotient to a linear function?
4. How does the result change if you modify the function to include a higher degree term?
5. How do you interpret the result of the difference quotient geometrically?

Tip: The difference quotient gives the average rate of change of a function over an interval. It becomes the derivative when the interval approaches zero.