If x+2y=2 and `x^(2)-y^(2)=9`, what is the value of `(x^(3)-2y^(3)-xy^(2)+2x^(2)y)` ?

Answer

To solve the problem, we need to find the value of the expression \( x^3 - 2y^3 - xy^2 + 2x^2y \) given the equations \( x + 2y = 2 \) and \( x^2 - y^2 = 9 \).

### Step-by-Step Solution:

1. **Identify the given equations:**
\[ x + 2y = 2 \quad \text{(1)} \]
\[ x^2 - y^2 = 9 \quad \text{(2)} \]

2. **Rearrange the expression:**
The expression we need to evaluate is:
\[ x^3 - 2y^3 - xy^2 + 2x^2y \]
We can group the terms:
\[ = (x^3 - xy^2) + (2x^2y - 2y^3) \]

3. **Factor the grouped terms:**
- From the first group \( x^3 - xy^2 \), we can factor out \( x \):
\[ = x(x^2 - y^2) \]
- From the second group \( 2x^2y - 2y^3 \), we can factor out \( 2y \):
\[ = 2y(x^2 - y^2) \]

4. **Combine the factored terms:**
Now we can write the expression as:
\[ = (x + 2y)(x^2 - y^2) \]

5. **Substitute the known values:**
From equation (1), we know \( x + 2y = 2 \).
From equation (2), we know \( x^2 - y^2 = 9 \).
Thus, we can substitute these values into our expression:
\[ = 2 \cdot 9 \]

6. **Calculate the final value:**
\[ = 18 \]

### Final Answer:
The value of the expression \( x^3 - 2y^3 - xy^2 + 2x^2y \) is \( \boxed{18} \).