Area of rectangle with length `x^2 + 2xy + y^2` and breadth `x^2- 2xy + y^2` is ________

Options

A. `x^4 + y^4 - 2x^2 y^2`
B. `x^2 + y^2 - 4x^2 y^2`
C. `x^4 + y^4 + 2x^2 y^2`
D. `x^4 - y^4 + 2x^2 y^2`

Answer

To find the area of a rectangle, we use the formula:

\[ \text{Area} = \text{Length} \times \text{Breadth} \]

Given:

- Length = \( x^2 + 2xy + y^2 \)
- Breadth = \( x^2 - 2xy + y^2 \)

### Step 1: Identify the expressions for length and breadth
The length is \( x^2 + 2xy + y^2 \) and the breadth is \( x^2 - 2xy + y^2 \).

### Step 2: Recognize the patterns in the expressions
The expression for length can be rewritten using the identity for a perfect square:

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

The expression for breadth can also be rewritten:

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

### Step 3: Substitute the identities into the area formula
Now substitute these identities into the area formula:

\[ \text{Area} = (x + y)^2 \times (x - y)^2 \]

### Step 4: Use the identity for the product of squares
We can use the identity \( a^2 \times b^2 = (a \times b)^2 \):

\[ \text{Area} = [(x + y)(x - y)]^2 \]

### Step 5: Simplify the expression
Now, simplify the expression:

\[ \text{Area} = (x^2 - y^2)^2 \]

### Final Answer
Thus, the area of the rectangle is:

\[ \text{Area} = (x^2 - y^2)^2 \]

---