The expression you provided is:

\[
\left(-\frac{X^5}{2} + Y\right) \left(4X^2 - XY + Y^3\right)
\]

Let’s expand this expression using the distributive property (FOIL):

\[
= \left(-\frac{X^5}{2}\right) \left(4X^2 - XY + Y^3\right) + Y \left(4X^2 - XY + Y^3\right)
\]

We’ll now expand each term separately:

1. First, distribute \(-\frac{X^5}{2}\) to each term inside the second parentheses:
   \[
   -\frac{X^5}{2} \cdot 4X^2 = -2X^7
   \]
   \[
   -\frac{X^5}{2} \cdot (-XY) = \frac{X^6Y}{2}
   \]
   \[
   -\frac{X^5}{2} \cdot Y^3 = -\frac{X^5Y^3}{2}
   \]

2. Now, distribute \(Y\) to each term inside the second parentheses:
   \[
   Y \cdot 4X^2 = 4X^2Y
   \]
   \[
   Y \cdot (-XY) = -X Y^2
   \]
   \[
   Y \cdot Y^3 = Y^4
   \]

Now combine all the terms together:

\[
-2X^7 + \frac{X^6Y}{2} - \frac{X^5Y^3}{2} + 4X^2Y - X Y^2 + Y^4
\]

This is the fully expanded form of the expression.

Would you like to go over any of the steps in more detail or have any further questions?

Here are 5 related questions you could explore:
1. How do you factor complex polynomials like this one?
2. Can you simplify the expression further under specific conditions?
3. What is the derivative of this expanded expression with respect to \(X\)?
4. How do terms involving powers of variables change during expansion?
5. How do you find critical points of this expression when treated as a function?

**Tip:** Always ensure you distribute each term in the parentheses correctly during expansion to avoid mistakes.

Learn how to expand the polynomial expression (-X^5/2 + Y)(4X^2 - XY + Y^3) using the distributive property. This solution includes detailed steps, from distributing terms to combining like terms. Suitable for students in Grades 10-12 studying polynomial expansion.

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