To find the perimeter of a rectangle when given the expressions for its two adjacent sides, we'll follow these steps in a detailed manner.
### Step 1: Identify the given expressions for the sides.
The two adjacent sides of the rectangle are given as:
1. [tex]\( a = 3x^2 - 2y^2 \)[/tex]
2. [tex]\( b = x^2 + 3xy \)[/tex]
### Step 2: Recall the formula for the perimeter of a rectangle.
For a rectangle, the perimeter [tex]\( P \)[/tex] is calculated using the formula:
[tex]\[ P = 2(a + b) \][/tex]
### Step 3: Substitute the given expressions into the formula.
Substituting [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula for the perimeter:
[tex]\[ P = 2((3x^2 - 2y^2) + (x^2 + 3xy)) \][/tex]
### Step 4: Simplify the expression inside the parentheses.
First, combine like terms:
[tex]\[ (3x^2 - 2y^2) + (x^2 + 3xy) = 3x^2 + x^2 - 2y^2 + 3xy = 4x^2 + 3xy - 2y^2 \][/tex]
### Step 5: Multiply by 2 to find the perimeter.
Finally, multiply the simplified expression by 2:
[tex]\[ P = 2(4x^2 + 3xy - 2y^2) \][/tex]
### Step 6: Distribute the 2 across the terms inside the parentheses.
[tex]\[ P = 2 \cdot 4x^2 + 2 \cdot 3xy - 2 \cdot 2y^2 \][/tex]
[tex]\[ P = 8x^2 + 6xy - 4y^2 \][/tex]
### Conclusion
Therefore, the perimeter of the rectangle, given the sides [tex]\( 3x^2 - 2y^2 \)[/tex] and [tex]\( x^2 + 3xy \)[/tex], is:
[tex]\[ P = 8x^2 + 6xy - 4y^2 \][/tex]
