Final answer:
The expression that represents the area of the pool, including the sidewalk, is 96 + 4x^2 + 56x square feet.
Step-by-step explanation:
The area of the rectangular pool, including the sidewalk, can be found by adding the areas of the pool and the sidewalk.
The area of the pool is determined by multiplying its length by its width. In this case, the length is 12 feet and the width is 8 feet, so the area of the pool is 12 feet * 8 feet = 96 square feet.
The area of the sidewalk is the difference between the area of the rectangle formed by the outer boundary of the pool and the rectangle formed by the inner boundary of the pool. The outer rectangle has length (12 + 2x) feet and width (8 + 2x) feet, while the inner rectangle has length 12 feet and width 8 feet. Therefore, the area of the sidewalk is (12 + 2x)*(8 + 2x) - 12 * 8 = (96 + 32x + 24x + 4x^2) - 96 = 4x^2 + 56x.
So, the expression that represents the area of the pool, including the sidewalk, is 96 + 4x^2 + 56x square feet.