The problem involves simplifying the difference quotient for the function $f(x) = 4x^2 + 5x - 1$. Specifically, you are tasked with calculating and simplifying:

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

Step-by-Step Solution:

1. Find $f(x+h)$:
   Substitute $x+h$ into the function $f(x) = 4x^2 + 5x - 1$.
   $f(x+h) = 4(x+h)^2 + 5(x+h) - 1.$ Expand $(x+h)^2$: $f(x+h) = 4(x^2 + 2xh + h^2) + 5x + 5h - 1.$ Simplify: $f(x+h) = 4x^2 + 8xh + 4h^2 + 5x + 5h - 1.$

2. Calculate $f(x+h) - f(x)$:
   Subtract $f(x) = 4x^2 + 5x - 1$ from $f(x+h)$:
   $f(x+h) - f(x) = \left(4x^2 + 8xh + 4h^2 + 5x + 5h - 1\right) - \left(4x^2 + 5x - 1\right).$ Cancel like terms: $f(x+h) - f(x) = 8xh + 4h^2 + 5h.$

3. Divide by $h$:
   Now divide the result by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{8xh + 4h^2 + 5h}{h}.$ Factor $h$ from the numerator: $\frac{f(x+h) - f(x)}{h} = \frac{h(8x + 4h + 5)}{h}.$ Cancel $h$ (assuming $h \neq 0$): $\frac{f(x+h) - f(x)}{h} = 8x + 4h + 5.$

Final Answer:

$\frac{f(x+h) - f(x)}{h} = 8x + 4h + 5.$

Would you like further clarifications on this or related problems?

Here are 5 related questions for practice:

1. How do you find the derivative of $f(x) = 4x^2 + 5x - 1$ using the limit definition of derivatives?
2. What happens to $\frac{f(x+h) - f(x)}{h}$ as $h \to 0$?
3. What is the significance of the difference quotient in calculus?
4. How does the function $f(x) = 4x^2 + 5x - 1$ behave (e.g., its graph and critical points)?
5. Expand and simplify $g(x+h)$ for a different function, such as $g(x) = 3x^3 - 2x$.

Tip: Always expand expressions carefully to avoid algebraic errors, especially when working with higher-degree polynomials.