The difference quotient is defined as:

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

where $f(x) = 3x^2 + 7x - 8$.

Let's calculate this step by step.

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

Substitute $x+h$ into the function $f(x)$:

$f(x+h) = 3(x+h)^2 + 7(x+h) - 8$

Expand the terms:

$f(x+h) = 3(x^2 + 2xh + h^2) + 7(x + h) - 8$ $f(x+h) = 3x^2 + 6xh + 3h^2 + 7x + 7h - 8$

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

Now subtract $f(x) = 3x^2 + 7x - 8$ from $f(x+h)$:

$f(x+h) - f(x) = \left( 3x^2 + 6xh + 3h^2 + 7x + 7h - 8 \right) - \left( 3x^2 + 7x - 8 \right)$

Simplify by canceling out common terms:

$f(x+h) - f(x) = 6xh + 3h^2 + 7h$

Step 3: Divide by $h$

Now, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 7h}{h}$

Simplify:

$\frac{f(x+h) - f(x)}{h} = 6x + 3h + 7$

Final Answer:

The difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 6x + 3h + 7$

Let me know if you'd like more details or have any questions!

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Here are 5 related questions:

1. How does the difference quotient relate to the derivative of a function?
2. What happens to the difference quotient as $h \to 0$?
3. How would you find the derivative of $f(x) = 3x^2 + 7x - 8$?
4. Can you calculate the difference quotient for a cubic function?
5. What is the geometric interpretation of the difference quotient?

Tip: The difference quotient is the foundation for finding derivatives, which describe the slope of a curve at any given point.