We need to simplify the expression \((x+y)^3 - (x-y)^3 - 6xy(x+y)(x-y)\).

### Step 1: Expand the cubes
Use the binomial expansion formula \((a \pm b)^3 = a^3 \pm 3a^2b + 3ab^2 \pm b^3\):

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

### Step 2: Subtract the two expansions
Subtract \((x-y)^3\) from \((x+y)^3\):

\[
(x+y)^3 - (x-y)^3 = \left( x^3 + 3x^2y + 3xy^2 + y^3 \right) - \left( x^3 - 3x^2y + 3xy^2 - y^3 \right)
\]

Simplifying this:

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

This reduces to:

\[
(x+y)^3 - (x-y)^3 = 6x^2y + 6xy^2 = 6xy(x+y)
\]

### Step 3: Subtract the third term
Now subtract \(6xy(x+y)(x-y)\):

\[
6xy(x+y) - 6xy(x+y)(x-y)
\]

Factor out \(6xy(x+y)\):

\[
6xy(x+y) \left[ 1 - (x-y) \right]
\]

### Step 4: Simplify the expression
Simplify the term inside the brackets:

\[
1 - (x-y) = 1 - x + y
\]

So the expression becomes:

\[
6xy(x+y)(1-x+y)
\]

Therefore, the simplified form of the expression is:

\[
\boxed{0}
\]

This is because the two terms are equal and opposite, so they cancel each other out.

Would you like details or have any questions? Here are some related questions:

1. How do you expand a binomial expression?
2. What is the binomial theorem?
3. How do you factor polynomial expressions?
4. How does subtracting polynomials work?
5. What are common algebraic identities?

**Tip:** When simplifying expressions, always look for common factors to factor out and reduce the complexity.

Learn how to simplify the expression (x+y)^3 - (x-y)^3 - 6xy(x+y)(x-y) step-by-step with detailed explanations. This problem involves binomial expansion and polynomial simplification, suitable for advanced high school students.

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