The function provided is $f(x) = -5x^2 - 3x + 8$. We need to evaluate two expressions:

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

Let's break this down.

1. Finding $f(x + h)$

To compute $f(x + h)$, substitute $x + h$ for $x$ in the function: $f(x + h) = -5(x + h)^2 - 3(x + h) + 8$ Now, expand this expression:

$(x + h)^2 = x^2 + 2xh + h^2$ So, $f(x + h) = -5(x^2 + 2xh + h^2) - 3(x + h) + 8$ Expanding further: $f(x + h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 8$

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

Now, subtract $f(x)$ from $f(x + h)$: $f(x + h) - f(x) = (-5x^2 - 10xh - 5h^2 - 3x - 3h + 8) - (-5x^2 - 3x + 8)$ Simplify this: $f(x + h) - f(x) = -10xh - 5h^2 - 3h$ Finally, divide 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 answers:

• $f(x + h) = -5x^2 - 10xh - 5h^2 - 3x - 3h + 8$
• $\frac{f(x + h) - f(x)}{h} = -10x - 5h - 3$

Would you like more details or clarifications on any of these steps?

Here are 5 related questions you might find useful:

1. How do you find the derivative of $f(x) = -5x^2 - 3x + 8$ using limits?
2. What happens as $h \to 0$ in the difference quotient $\frac{f(x+h) - f(x)}{h}$?
3. How do you find the tangent line to $f(x)$ at a given point?
4. What is the significance of the difference quotient in calculus?
5. How does expanding $(x + h)^2$ help in simplifying the difference quotient?

Tip: The difference quotient $\frac{f(x + h) - f(x)}{h}$ is the foundation for finding the derivative, which represents the slope of the tangent line to the function at any point $x$.