Find each of the following products :
`(x^(2) + xy + y^(2)) xx (x - y)`

Answer

To solve the problem of finding the product \((x^2 + xy + y^2)(x - y)\), we will use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step breakdown:

### Step 1: Distribute the first term of the first expression
Multiply \(x^2\) by each term in the second expression \((x - y)\):
\[ x^2 \cdot x - x^2 \cdot y = x^3 - x^2y \]

### Step 2: Distribute the second term of the first expression
Now, multiply \(xy\) by each term in the second expression \((x - y)\):
\[ xy \cdot x - xy \cdot y = x^2y - y^2x \]

### Step 3: Distribute the third term of the first expression
Finally, multiply \(y^2\) by each term in the second expression \((x - y)\):
\[ y^2 \cdot x - y^2 \cdot y = y^2x - y^3 \]

### Step 4: Combine all the terms
Now, we will combine all the results from Steps 1, 2, and 3:
\[ (x^3 - x^2y) + (x^2y - y^2x) + (y^2x - y^3) \]

### Step 5: Simplify the expression
Now, combine like terms:
- The \(x^2y\) and \(-x^2y\) cancel each other out.
- The \(-y^2x\) and \(y^2x\) also cancel each other out.

So, we are left with:
\[ x^3 - y^3 \]

### Final Answer
Thus, the product \((x^2 + xy + y^2)(x - y)\) simplifies to:
\[ \boxed{x^3 - y^3} \]

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