To find the perimeter and area of the polygon, let's assume the polygon is a rectangle, with the given expressions representing the lengths of its sides.
Given:
- One side has a length of [tex]\(3x - 4\)[/tex]
- The adjacent side has a length of [tex]\(5x + 1\)[/tex]
### Finding the Perimeter
The perimeter [tex]\(P\)[/tex] of a rectangle is calculated as:
[tex]\[ P = 2 \times (\text{length} + \text{width}) \][/tex]
Here, the length is [tex]\(3x - 4\)[/tex] and the width is [tex]\(5x + 1\)[/tex]:
1. Add the expressions for the length and width:
[tex]\[
(3x - 4) + (5x + 1) = 3x + 5x - 4 + 1 = 8x - 3
\][/tex]
2. Multiply the result by 2 to get the perimeter:
[tex]\[
P = 2 \times (8x - 3) = 16x - 6
\][/tex]
So, the perimeter of the rectangle is:
[tex]\[ \boxed{16x - 6} \][/tex]
### Finding the Area
The area [tex]\(A\)[/tex] of a rectangle is calculated as:
[tex]\[ A = \text{length} \times \text{width} \][/tex]
Given the length [tex]\(3x - 4\)[/tex] and the width [tex]\(5x + 1\)[/tex], the area is:
[tex]\[
A = (3x - 4) \times (5x + 1)
\][/tex]
To find the product, use the distributive property (FOIL method):
[tex]\[
A = (3x)(5x) + (3x)(1) - (4)(5x) - (4)(1)
\][/tex]
Simplify each term:
[tex]\[
A = 15x^2 + 3x - 20x - 4
\][/tex]
Combine like terms:
[tex]\[
A = 15x^2 - 17x - 4
\][/tex]
So, the area of the rectangle is:
[tex]\[ \boxed{15x^2 - 17x - 4} \][/tex]
To summarize:
- The perimeter of the polygon (assumed to be a rectangle) is [tex]\(16x - 6\)[/tex].
- The area of the polygon (assumed to be a rectangle) is [tex]\(15x^2 - 17x - 4\)[/tex].
