The given expression is:

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

Let's simplify it step by step:

### Step 1: Expand the first term \( x(x - y)^3 \)
Leave the cubic term as it is for now:
\[
x(x - y)^3
\]
This can be written as:
\[
x \cdot (x - y)^3
\]

### Step 2: Expand the second term \( -3xy(x + y) \)
Distribute \( -3xy \) over \( (x + y) \):
\[
-3xy(x + y) = -3xy \cdot x - 3xy \cdot y
\]
Simplifying each term:
\[
-3xy \cdot x = -3x^2y
\]
\[
-3xy \cdot y = -3xy^2
\]
So:
\[
-3xy(x + y) = -3x^2y - 3xy^2
\]

### Step 3: Combine the terms
Now, the expression becomes:
\[
x(x - y)^3 - 3x^2y - 3xy^2
\]

### Final Answer:
Thus, the simplified expression is:
\[
x(x - y)^3 - 3x^2y - 3xy^2
\]

Would you like to expand further, or do you have any specific questions on this? Let me know if you'd like more details.

Here are 5 related questions to deepen understanding:
1. How would you expand \( (x - y)^3 \)?
2. What is the significance of distributing terms in algebra?
3. Can you factor \( x(x - y)^3 \) in any other way?
4. What happens if you set \( y = 0 \) in the original expression?
5. How would the expression change if the exponent on \( (x - y) \) were different?

**Tip:** When simplifying algebraic expressions, it's often useful to expand carefully, checking for common terms to combine or factor.

Discover how to simplify the algebraic expression x(x-y)^3 - 3xy(x+y) using polynomial expansion and the distributive property. Learn key algebraic techniques such as expanding cubic terms and simplifying polynomial expressions. Suitable for high school students in Grades 9-12.

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