To evaluate and simplify $\frac{f(x+h) - f(x)}{h}$ for the function $f(x) = 2x^2 - x$, let's follow these steps:

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

   $f(x+h) = 2(x+h)^2 - (x+h)$

   Expanding $(x+h)^2$:

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

   Therefore,

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

2. Find $f(x+h) - f(x)$:

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

   $f(x+h) - f(x) = \left(2x^2 + 4xh + 2h^2 - x - h\right) - \left(2x^2 - x\right)$

   Simplifying:

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

3. Divide by $h$:

   Finally, divide by $h$:

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

   Simplifying by dividing each term by $h$:

   $\frac{f(x+h) - f(x)}{h} = 4x + 2h - 1$

The correct answer is $\boxed{4x + 2h - 1}$.

Would you like more details on this, or do you have any other questions?

Here are five related questions:

1. How do you find the derivative of $f(x)$ using the limit definition?
2. Can you apply the same method to find the difference quotient for $f(x) = 3x^2 - 4x + 1$?
3. What is the significance of the difference quotient in calculus?
4. How would you interpret the result $4x + 2h - 1$ in terms of a slope?
5. How does the expression change if $h$ approaches zero?

Tip: Understanding the difference quotient is essential, as it forms the basis for finding derivatives in calculus.