To evaluate the difference quotient for the function $f(x) = 4x^2 - 3x + 5$, we use the formula for the difference quotient:

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

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

Substitute $x+h$ into the function $f(x) = 4x^2 - 3x + 5$:

$f(x+h) = 4(x+h)^2 - 3(x+h) + 5$

First, expand $(x+h)^2$:

$(x+h)^2 = x^2 + 2xh + h^2$

Now substitute this back into the function:

$f(x+h) = 4(x^2 + 2xh + h^2) - 3(x+h) + 5$

Expand the terms:

$f(x+h) = 4x^2 + 8xh + 4h^2 - 3x - 3h + 5$

Step 2: Calculate the Difference Quotient

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

$f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 - 3x - 3h + 5) - (4x^2 - 3x + 5)$

Simplify the expression:

$f(x+h) - f(x) = 8xh + 4h^2 - 3h$

Now, divide the result by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 - 3h}{h}$

Simplify by canceling out $h$ from each term:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3$

Final Answer:

The difference quotient is:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h - 3$

Do you want more details on this process or have any questions?

Here are 5 related questions for further exploration:

1. What happens to the difference quotient as $h$ approaches 0?
2. How do you interpret the difference quotient geometrically?
3. How would the difference quotient change if the function had a cubic term?
4. What is the derivative of $f(x) = 4x^2 - 3x + 5$ using limits?
5. Can you evaluate the difference quotient for $f(x) = x^3 + 2x$?

Tip: The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval.