To determine the correct expression for the perimeter of rectangle [tex]\( A \)[/tex] and to verify the closure property, follow these steps:
1. Identify the Expressions for Length and Width:
- The length of rectangle [tex]\( A \)[/tex] is given as [tex]\( 3x + 5 \)[/tex].
- The width of rectangle [tex]\( A \)[/tex] is given as [tex]\( 2x - 3 \)[/tex].
2. Formula for Perimeter of a Rectangle:
The formula to calculate the perimeter [tex]\( P \)[/tex] of a rectangle is:
[tex]\[
P = 2 \times (\text{Length} + \text{Width})
\][/tex]
3. Substitute the Given Length and Width:
Substitute the given expressions for length and width into the perimeter formula:
[tex]\[
P = 2 \times ( (3x + 5) + (2x - 3) )
\][/tex]
4. Simplify the Expression Inside the Parentheses:
Combine like terms inside the parentheses:
[tex]\[
(3x + 5) + (2x - 3) = 3x + 2x + 5 - 3 = 5x + 2
\][/tex]
5. Calculate the Perimeter:
Multiply by 2 to get the expression for the perimeter:
[tex]\[
P = 2 \times (5x + 2) = 2 \times 5x + 2 \times 2 = 10x + 4
\][/tex]
6. Analyze the Resulting Expression:
The resulting expression for the perimeter of rectangle [tex]\( A \)[/tex] is [tex]\( 10x + 4 \)[/tex].
7. Check the Closure Property:
The closure property states that performing arithmetic operations (addition, subtraction, multiplication, or division by a non-zero number) on polynomials will result in another polynomial. Here, [tex]\( 10x + 4 \)[/tex] is a polynomial because it is a linear expression involving terms with non-negative integer exponents and does not involve any division operations.
Therefore, the correct expression for the perimeter of rectangle [tex]\( A \)[/tex] is [tex]\( 10x + 4 \)[/tex], and it is a polynomial.
[tex]\[
\boxed{10x + 4; \text{ the answer is a polynomial}}
\][/tex]
