You’ve provided the function:

$f(x) = x^3 + 3x^2 - 24x - 80$

and the condition $x_0 = -4$.

Could you clarify what you need help with? Are you trying to evaluate the function at $x = -4$, find the derivative, or something else like solving for the roots or finding specific values based on this condition?

Feel free to specify, and I'll be glad to assist you!

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Meanwhile, here are some related questions you might find helpful:

1. How do you evaluate a polynomial at a given point?
2. What is the derivative of $f(x)$ and how does it relate to the function's graph?
3. How do you solve for the roots of a cubic equation?
4. What is the significance of $x_0 = -4$ in this context (if it's part of a limit, derivative, or other condition)?
5. How can you determine the behavior of $f(x)$ at critical points?

Tip: To evaluate a polynomial function at a given point, simply substitute that value for $x$ into the expression and perform the arithmetic operations.