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

Answer

To find the perimeter of a rectangle given its adjacent sides, we can follow these steps:

### Step 1: Identify the Length and Breadth
The two adjacent sides of the rectangle are given as:
- Length \( L = 5x^2 - 3y^2 \)
- Breadth \( B = x^2 + 2xy \)

### Step 2: Write the Formula for Perimeter
The formula for the perimeter \( P \) of a rectangle is:
\[ P = 2 \times (L + B) \]

### Step 3: Substitute the Values of Length and Breadth
Now, substitute the values of \( L \) and \( B \) into the perimeter formula:
\[ P = 2 \times \left( (5x^2 - 3y^2) + (x^2 + 2xy) \right) \]

### Step 4: Simplify the Expression Inside the Parentheses
Combine like terms:
\[ L + B = (5x^2 + x^2) + (2xy) - 3y^2 = 6x^2 + 2xy - 3y^2 \]

### Step 5: Multiply by 2 to Find the Perimeter
Now, multiply the simplified expression by 2:
\[ P = 2 \times (6x^2 + 2xy - 3y^2) \]
Distributing the 2:
\[ P = 12x^2 + 4xy - 6y^2 \]

### Final Answer
Thus, the perimeter of the rectangle is:
\[ P = 12x^2 + 4xy - 6y^2 \]

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