The expression you've provided is:

\[
-y(4x^2 - 3x^2y + xy^2 + 7y^3)
\]

Now, let's distribute the \(-y\) across the terms inside the parentheses:

1. \(-y \times 4x^2 = -4x^2y\)
2. \(-y \times (-3x^2y) = +3x^2y^2\)
3. \(-y \times (xy^2) = -xy^3\)
4. \(-y \times (7y^3) = -7y^4\)

Thus, the expanded expression is:

\[
-4x^2y + 3x^2y^2 - xy^3 - 7y^4
\]

Would you like more details on the distribution process or have any questions?

Here are five related questions for practice:
1. How would the expression change if \( -y \) were instead \( +y \)?
2. What happens if you substitute specific values for \(x\) and \(y\)?
3. Can you factor this expression?
4. How does this expression behave as \(y\) approaches 0?
5. What if you differentiate this expression with respect to \(x\)?

**Tip:** When distributing a negative sign, be sure to apply it to all terms inside the parentheses.

Learn how to distribute and simplify the expression -y(4x^2 - 3x^2y + xy^2 + 7y^3). This problem covers important algebraic concepts, including polynomial expansion and the distributive property. Suitable for students in Grades 9-12.

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