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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?

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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.

Compute and simplify the difference quotient (f(x+h) - f(x)) / h for the function f(x) = 5x^2 - 3x - 7.

Learn how to compute and simplify the difference quotient for the quadratic function f(x) = 5x² - 3x - 7. This solution includes expanding binomials, simplifying expressions, and dividing by h. Essential for understanding the foundation of calculus.

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