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

Answer

To solve the problem `(x^(2) - y^(2)) * (x + 2y)`, we will use the distributive property to multiply each term in the first expression by each term in the second expression.

### Step-by-Step Solution:

1. **Identify the Terms:**
We have two expressions:
- First expression: \( x^2 - y^2 \)
- Second expression: \( x + 2y \)

2. **Distribute the First Term:**
We will first multiply \( x^2 \) by each term in the second expression:
\[ x^2 \cdot (x + 2y) = x^2 \cdot x + x^2 \cdot 2y \]
- \( x^2 \cdot x = x^{2+1} = x^3 \)
- \( x^2 \cdot 2y = 2x^2y \)

So, from this step, we get:
\[ x^3 + 2x^2y \]

3. **Distribute the Second Term:**
Next, we will multiply \( -y^2 \) by each term in the second expression:
\[ -y^2 \cdot (x + 2y) = -y^2 \cdot x + (-y^2) \cdot 2y \]
- \( -y^2 \cdot x = -xy^2 \)
- \( -y^2 \cdot 2y = -2y^{2+1} = -2y^3 \)

So, from this step, we get:
\[ -xy^2 - 2y^3 \]

4. **Combine All Terms:**
Now, we combine all the terms obtained from both distributions:
\[ x^3 + 2x^2y - xy^2 - 2y^3 \]

5. **Final Answer:**
The final product of the expression \( (x^2 - y^2)(x + 2y) \) is:
\[ x^3 + 2x^2y - xy^2 - 2y^3 \]