The product of the rational expressions `(x^2 - y^2)/(x^2 + 2xy + y^2)` and `(xy + y^2)/(x^2 - xy)` is:

Options

A. `xy`
B. `y//x`
C. `x//y`
D. `1`

Answer

To solve the problem, we need to calculate the product of the rational expressions \((x^2 - y^2)/(x^2 + 2xy + y^2)\) and \((xy + y^2)/(x^2 - xy)\). Let's go through the solution step by step.

### Step 1: Factor the Numerators and Denominators
First, we will factor the expressions in the numerators and denominators.

1. **Numerator of the first expression**:
\[ x^2 - y^2 = (x + y)(x - y) \quad \text{(Difference of squares)} \]

2. **Denominator of the first expression**:
\[ x^2 + 2xy + y^2 = (x + y)^2 \quad \text{(Perfect square trinomial)} \]

3. **Numerator of the second expression**:
\[ xy + y^2 = y(x + y) \quad \text{(Factoring out y)} \]

4. **Denominator of the second expression**:
\[ x^2 - xy = x(x - y) \quad \text{(Factoring out x)} \]

### Step 2: Rewrite the Product
Now we can rewrite the product of the two rational expressions using the factored forms:

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

### Step 3: Combine the Expressions
Now combine the fractions:

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

### Step 4: Simplify the Expression
Now we can simplify the expression:

1. **Cancel out** \((x + y)\) from the numerator and denominator:
\[ = \frac{(x - y) \cdot y}{(x + y) \cdot x} \]

2. **Notice that** \((x - y)\) cancels out:
\[ = \frac{y}{x} \]

### Final Answer
Thus, the product of the rational expressions is:

\[ \frac{y}{x} \]