The adjacent sides of a rectangle are `3x^(2) - 2xy + 5y^(2)` and `2x^(2) + 5xy - 3y^(2)`. Find the area of the rectangle.

Answer

To find the area of the rectangle with given adjacent sides, we will follow these steps:

### Step 1: Identify the Length and Breadth
Let the length \( L \) and breadth \( B \) of the rectangle be defined as follows:
- \( L = 3x^2 - 2xy + 5y^2 \)
- \( B = 2x^2 + 5xy - 3y^2 \)

### Step 2: Use the Area Formula
The area \( A \) of a rectangle is given by the formula:
\[ A = L \times B \]
Substituting the values of \( L \) and \( B \):
\[ A = (3x^2 - 2xy + 5y^2)(2x^2 + 5xy - 3y^2) \]

### Step 3: Expand the Expression
We will expand the expression using the distributive property (also known as the FOIL method for binomials):
\[ A = (3x^2)(2x^2) + (3x^2)(5xy) + (3x^2)(-3y^2) + (-2xy)(2x^2) + (-2xy)(5xy) + (-2xy)(-3y^2) + (5y^2)(2x^2) + (5y^2)(5xy) + (5y^2)(-3y^2) \]

Calculating each term:
1. \( 3x^2 \cdot 2x^2 = 6x^4 \)
2. \( 3x^2 \cdot 5xy = 15x^3y \)
3. \( 3x^2 \cdot -3y^2 = -9x^2y^2 \)
4. \( -2xy \cdot 2x^2 = -4x^3y \)
5. \( -2xy \cdot 5xy = -10x^2y^2 \)
6. \( -2xy \cdot -3y^2 = 6xy^3 \)
7. \( 5y^2 \cdot 2x^2 = 10x^2y^2 \)
8. \( 5y^2 \cdot 5xy = 25xy^3 \)
9. \( 5y^2 \cdot -3y^2 = -15y^4 \)

### Step 4: Combine Like Terms
Now, we will combine all the like terms:
- \( 6x^4 \)
- \( 15x^3y - 4x^3y = 11x^3y \)
- \( -9x^2y^2 - 10x^2y^2 + 10x^2y^2 = -9x^2y^2 \)
- \( 6xy^3 + 25xy^3 = 31xy^3 \)
- \( -15y^4 \)

Putting it all together, we have:
\[ A = 6x^4 + 11x^3y - 9x^2y^2 + 31xy^3 - 15y^4 \]

### Final Answer
The area of the rectangle is:
\[ A = 6x^4 + 11x^3y - 9x^2y^2 + 31xy^3 - 15y^4 \]