The difference quotient formula for a function $f(x)$ is given by:

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

For $f(x) = 4x - x^2$, let's compute $f(x+h)$ first:

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

Now, let's simplify $f(x+h)$:

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

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

Now, substitute $f(x+h)$ and $f(x) = 4x - x^2$ into the difference quotient formula:

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

Simplify the expression:

$= \frac{4x + 4h - x^2 - 2xh - h^2 - 4x + x^2}{h}$

Cancel out terms:

$= \frac{4h - 2xh - h^2}{h}$

Now, divide each term by $h$:

$= 4 - 2x - h$

Thus, the difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 4 - 2x - h$

Would you like further clarification or details?

Here are 5 related questions to expand on this topic:

1. How does the difference quotient relate to the derivative of a function?
2. Can you find the derivative of $f(x) = 4x - x^2$ using first principles?
3. What happens to the difference quotient as $h$ approaches 0?
4. How do you apply the difference quotient to more complex functions?
5. Can you explain how to interpret the slope of a function geometrically?

Tip: The difference quotient is the foundation for understanding derivatives in calculus, which measure the instantaneous rate of change of a function.