It may make more sense if you sketch it out. First draw the 23 x 47 pool. Then draw another rectangle around that where the distance between the edge of the pool and the edge of the walk (or the width of the walk) is an unknown, but uniform, length. We'll call this x.
If we take the area of the larger rectangle and subtract out the area of the inner rectangle, we have the area of the path.
Area of the pool is easy:
23 * 47 = 1081 ft²
The area of the complete larger area contains your unknown dimensions.
If you look at it, it's 23 plus two x's for one side and 47 plus two x's for the other.
Multiply those together to get your area.
(23 + 2x)(47 + 2x)
4x² + 140x + 1081 ft²
Now if we subtract out the area of the pool from the area of the entire rectangle,
we have the area of the walkway:
1081 + 140x + 4x² - 1081 = 4x² + 140x ft²
And we know this area is 456 ft², so now we can set this expression to that value, and solve for x:
4x² + 140x = 456
divide both sides by 4:
x² + 35x = 114
x² + 35x - 114 = 0
(x + 39)(x - 3) = 0
x = -39 and 3
Since we can't have a negative distance, we'll throw that out and we are left with:
x = 3
The width of the walk all the way around the pool is 3 ft.