Let's simplify the expression:

\[
-3xy^6(5y^5 - 4x^2 + 8)
\]

Distribute \(-3xy^6\) across each term inside the parentheses:

1. Multiply \(-3xy^6\) by \(5y^5\):
   \[
   -3xy^6 \times 5y^5 = -15xy^{11}
   \]

2. Multiply \(-3xy^6\) by \(-4x^2\):
   \[
   -3xy^6 \times -4x^2 = 12x^3y^6
   \]

3. Multiply \(-3xy^6\) by \(8\):
   \[
   -3xy^6 \times 8 = -24xy^6
   \]

Putting it all together, the simplified expression is:

\[
-15xy^{11} + 12x^3y^6 - 24xy^6
\]

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

Here are five related questions you might find interesting:

1. How would the expression change if the \(y^6\) factor was factored out from the entire expression?
2. What happens if you substitute specific values for \(x\) and \(y\) in the simplified expression?
3. How does the distribution change if you start with a positive coefficient instead of \(-3xy^6\)?
4. How would the expression simplify if the term inside the parentheses had only one term, like \(7y^5\)?
5. How would you factor the expression if it was presented in a different order?

**Tip:** When distributing a term over a polynomial, make sure to multiply both the coefficients and the variables with their respective exponents carefully.

Learn how to simplify the expression -3xy^6(5y^5 - 4x^2 + 8) step-by-step with detailed explanations. This problem covers polynomial multiplication and distribution. Suitable for students in Grades 7-9.

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