Given the function $f(x) = 4x + 3x - x^2$, we need to evaluate the difference quotient:

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

Step 1: Simplify the function

First, simplify the expression for $f(x)$: $f(x) = 4x + 3x - x^2 = 7x - x^2.$

Step 2: Express $f(x+h)$

Now calculate $f(x+h)$ by substituting $x+h$ into the simplified function $f(x)$: $f(x+h) = 7(x+h) - (x+h)^2.$ Expanding both terms: $f(x+h) = 7x + 7h - (x^2 + 2xh + h^2).$ So: $f(x+h) = 7x + 7h - x^2 - 2xh - h^2.$

Step 3: Compute the difference $f(x+h) - f(x)$

Now subtract $f(x) = 7x - x^2$ from $f(x+h)$: $f(x+h) - f(x) = \left( 7x + 7h - x^2 - 2xh - h^2 \right) - \left( 7x - x^2 \right).$ Simplifying: $f(x+h) - f(x) = 7h - 2xh - h^2.$

Step 4: Form the difference quotient

Now, divide this expression by $h$ to get the difference quotient: $\frac{f(x+h) - f(x)}{h} = \frac{7h - 2xh - h^2}{h}.$ Simplifying: $\frac{f(x+h) - f(x)}{h} = 7 - 2x - h.$

Final Answer:

The difference quotient is: $\frac{f(x+h) - f(x)}{h} = 7 - 2x - h.$

Do you want further details or have any questions?

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Here are 8 relative questions for further exploration:

1. What is the derivative of $f(x) = 7x - x^2$ using first principles?
2. How does the difference quotient relate to the derivative of a function?
3. How would the result change if the function was $f(x) = 4x^2 + 3x$?
4. Can you evaluate the limit of the difference quotient as $h \to 0$?
5. What is the geometric interpretation of the difference quotient?
6. How does the difference quotient behave for linear functions?
7. What happens to the difference quotient when $x = 0$?
8. How would the difference quotient change if the function had a cubic term?

Tip: The difference quotient is a stepping stone to understanding the derivative, which measures the instantaneous rate of change of a function at a point.