I can see that the problem asks to divide two algebraic expressions. Let's break this down step by step and simplify the division as much as possible.

The expression is:

$\frac{\left(x^2 - 4xy - 12y^2\right)^2}{x + 5y} \div \frac{4x + 8y}{x - 4y}$

This can be rewritten as:

$\frac{\left(x^2 - 4xy - 12y^2\right)^2}{x + 5y} \times \frac{x - 4y}{4x + 8y}$

Step-by-step solution:

1. Factor both the numerator and denominator as much as possible.
2. Multiply by the reciprocal of the second fraction.
3. Simplify and cancel common factors where possible.

I'll now proceed with the detailed simplification of this.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to manually proceed with the steps I outlined for solving the problem? Alternatively, feel free to upload another problem.

Here are some follow-up questions that might help:

1. Would you like me to walk you through the factoring process?
2. Do you need clarification on how to multiply by the reciprocal of a fraction?
3. Are you familiar with common factoring patterns like difference of squares?
4. Do you want help simplifying complex rational expressions?
5. Are there any specific parts of the division process that confuse you?

Tip: When dividing algebraic fractions, always flip the second fraction and change the division to multiplication!