Simplify the expression :
`[(x^3 + y^3)/((x - y)^2 + 3xy)] div [((x + y)^2 - 3xy)/(x^3 - y^3)] xx (xy)/(x^2 - y^2)`

Answer

To simplify the expression

\[ \frac{x^3 + y^3}{(x - y)^2 + 3xy} \div \frac{(x + y)^2 - 3xy}{x^3 - y^3} \cdot \frac{xy}{x^2 - y^2} \]

we will follow these steps:

### Step 1: Simplify \(x^3 + y^3\)

Using the identity for the sum of cubes, we have:

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

### Step 2: Simplify \((x - y)^2 + 3xy\)

Expanding \((x - y)^2\):

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

Thus,

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

### Step 3: Simplify \((x + y)^2 - 3xy\)

Using the expansion for \((x + y)^2\):

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

So,

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

### Step 4: Simplify \(x^3 - y^3\)

Using the identity for the difference of cubes, we have:

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

### Step 5: Simplify \(x^2 - y^2\)

Using the difference of squares, we get:

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

### Step 6: Substitute back into the original expression

Now substituting everything back into the original expression, we have:

\[ \frac{(x + y)(x^2 - xy + y^2)}{x^2 + y^2 + xy} \div \frac{(x^2 + y^2 - xy)}{(x - y)(x^2 + xy + y^2)} \cdot \frac{xy}{(x - y)(x + y)} \]

### Step 7: Change division to multiplication

Changing the division to multiplication by taking the reciprocal:

\[ = \frac{(x + y)(x^2 - xy + y^2)}{x^2 + y^2 + xy} \cdot \frac{(x - y)(x^2 + xy + y^2)}{(x^2 + y^2 - xy)} \cdot \frac{xy}{(x - y)(x + y)} \]

### Step 8: Cancel out common terms

Now we can cancel out the common terms:

- \(x + y\) in the numerator and denominator
- \(x - y\) in the numerator and denominator

This simplifies to:

\[ = \frac{(x^2 - xy + y^2) \cdot xy}{x^2 + y^2 + xy} \cdot \frac{(x^2 + xy + y^2)}{(x^2 + y^2 - xy)} \]

### Step 9: Final simplification

After canceling and simplifying, we are left with:

\[ = xy \]

### Final Answer:

Thus, the simplified expression is:

\[ \boxed{xy} \]