Let's compute the difference quotient for the given function:

The function is: $f(x) = 5x^2 - 3x - 7$

The difference quotient is defined as: $\frac{f(x+h) - f(x)}{h}$

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

We substitute $x+h$ into the function: $f(x+h) = 5(x+h)^2 - 3(x+h) - 7$ Expand $(x+h)^2$: $(x+h)^2 = x^2 + 2xh + h^2$ Thus: $f(x+h) = 5(x^2 + 2xh + h^2) - 3(x+h) - 7$ Expanding: $f(x+h) = 5x^2 + 10xh + 5h^2 - 3x - 3h - 7$

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

Now, subtract $f(x)$ from $f(x+h)$: $f(x+h) - f(x) = \left( 5x^2 + 10xh + 5h^2 - 3x - 3h - 7 \right) - \left( 5x^2 - 3x - 7 \right)$ Simplifying: $f(x+h) - f(x) = 5x^2 + 10xh + 5h^2 - 3x - 3h - 7 - 5x^2 + 3x + 7$ Canceling out terms: $f(x+h) - f(x) = 10xh + 5h^2 - 3h$

Step 3: Divide by $h$

Now, divide the result by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{10xh + 5h^2 - 3h}{h}$ Simplify: $\frac{f(x+h) - f(x)}{h} = 10x + 5h - 3$

Final Answer:

The simplified difference quotient is: $\boxed{10x + 5h - 3}$

Would you like further details or clarifications?

Here are 5 related questions to expand on this concept:

1. How is the difference quotient related to the derivative?
2. What happens to the expression when $h \to 0$?
3. How would the difference quotient change if the function were cubic, for example, $f(x) = 2x^3 - 5x + 1$?
4. Can you compute the derivative of the given function $f(x) = 5x^2 - 3x - 7$?
5. How does the difference quotient represent the average rate of change?

Tip: The difference quotient formula is a foundational concept for understanding derivatives, which measure the instantaneous rate of change of a function.