The two adjacent sides of a rectangle are `2x^(2) - 5xy + 3z^(2)` and `4xy - x^(2) - z^(2)`. Find its perimeter.

Answer

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

### Step 1: Identify the lengths of the sides
The two adjacent sides of the rectangle are given as:
- Length (AB) = \(2x^2 - 5xy + 3z^2\)
- Breadth (AD) = \(4xy - x^2 - z^2\)

### Step 2: Write the formula for the perimeter of a rectangle
The perimeter \(P\) of a rectangle is given by the formula:
\[ P = 2 \times (\text{Length} + \text{Breadth}) \]

### Step 3: Substitute the values of length and breadth into the formula
Substituting the expressions for length and breadth into the perimeter formula:
\[ P = 2 \times \left((2x^2 - 5xy + 3z^2) + (4xy - x^2 - z^2)\right) \]

### Step 4: Simplify the expression inside the parentheses
Now, combine the like terms:
\[ P = 2 \times \left(2x^2 - x^2 - 5xy + 4xy + 3z^2 - z^2\right) \]
This simplifies to:
\[ P = 2 \times \left((2x^2 - x^2) + (-5xy + 4xy) + (3z^2 - z^2)\right) \]
\[ P = 2 \times \left(x^2 - xy + 2z^2\right) \]

### Step 5: Distribute the 2
Now, distribute the 2:
\[ P = 2x^2 - 2xy + 4z^2 \]

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

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