The problem in the image asks to find and simplify the difference quotient for the given function:

$f(x) = -x^2 + 7x + 4$

The difference quotient formula is:

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

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

We need to plug $x+h$ into the function $f(x)$:

$f(x+h) = -(x+h)^2 + 7(x+h) + 4$

Now expand this expression:

$f(x+h) = -(x^2 + 2xh + h^2) + 7(x+h) + 4$ $= -x^2 - 2xh - h^2 + 7x + 7h + 4$

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

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

$f(x+h) - f(x) = (-x^2 - 2xh - h^2 + 7x + 7h + 4) - (-x^2 + 7x + 4)$ $= -x^2 - 2xh - h^2 + 7x + 7h + 4 + x^2 - 7x - 4$ Simplify: $= -2xh - h^2 + 7h$

Step 3: Divide by $h$

Now divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2 + 7h}{h}$

Simplify each term:

$= -2x - h + 7$

Final Result:

$\frac{f(x+h) - f(x)}{h} = -2x - h + 7$

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:

1. How is the difference quotient related to the derivative of a function?
2. What happens to the difference quotient as $h$ approaches 0?
3. Why do we expand $f(x+h)$ before subtracting $f(x)$?
4. Can you compute the derivative of $f(x) = -x^2 + 7x + 4$ using the power rule and compare it with the difference quotient?
5. How would the difference quotient change if we had a cubic function instead?

Tip: The difference quotient gives an approximation of the derivative, which represents the slope of the tangent line at any point on the curve.