Question

A rectangular swimming pool with length 50 ft and width 30 ft is surrounded by a sidewalk of uniform width. write a polynomial function that describes the area of the sidewalk in terms of its width

Answer 1

Let x represent the width of the sidewalk.
the swimming pool is a rectangle, l=50, w=30
the sidewalk forms a larger rectangle, l=50+2x, width=30+2x 
to find the area of the sidewalk, you can subtract the area of larger rectangle from the smaller rectange:
(50+2x)(30+2x)-(50*30)=4x²+160x

Another to do it is the cut sidewalk into 4 portions by extending the vertical sides of the pool to the outer edge of the sidewalk.
the two rectangle on the left and right each has area of (30+2x)*x=2x²+30x
the two horizontal rectangle each has an area: 50x
add them up: 2x²+30x+2x²+30x+50x+50x=4x²+160x