The problem asks you to evaluate two expressions for the given function $f(x) = -4x^2 - x - 1$.

First Expression: $f(x + h)$

To find $f(x + h)$, substitute $x + h$ into the function for $x$:

$f(x + h) = -4(x + h)^2 - (x + h) - 1$

Now, expand and simplify:

1. Expand $(x + h)^2$: $(x + h)^2 = x^2 + 2xh + h^2$

2. Substitute this back into the function: $f(x + h) = -4(x^2 + 2xh + h^2) - (x + h) - 1$

3. Distribute the constants: $f(x + h) = -4x^2 - 8xh - 4h^2 - x - h - 1$

Thus, the simplified form of $f(x + h)$ is: $f(x + h) = -4x^2 - 8xh - 4h^2 - x - h - 1$

Second Expression: $\frac{f(x + h) - f(x)}{h}$

Now, use the expressions for $f(x + h)$ and $f(x)$ to find this difference quotient:

$\frac{f(x + h) - f(x)}{h} = \frac{(-4x^2 - 8xh - 4h^2 - x - h - 1) - (-4x^2 - x - 1)}{h}$

Simplify the numerator: $= (-4x^2 - 8xh - 4h^2 - x - h - 1) + 4x^2 + x + 1$

Cancel the like terms: $= (-8xh - 4h^2 - h)$

So, the numerator simplifies to: $-8xh - 4h^2 - h$

Now divide by $h$: $\frac{-8xh - 4h^2 - h}{h} = -8x - 4h - 1$

Thus, the simplified form of the difference quotient is: $\frac{f(x + h) - f(x)}{h} = -8x - 4h - 1$

Would you like more details on any part of the solution or clarification on a step? Here are some related questions you might find useful:

1. What is the geometric meaning of the difference quotient?
2. How does this function represent a parabola?
3. How would the result change if the function had a different degree?
4. Can you apply this method to a cubic function?
5. What happens to the difference quotient as $h$ approaches 0?

Tip: The difference quotient is the foundation for finding the derivative of a function in calculus!