You don't understand because the person who gave you this answer is doing so many steps in his head that he is confusing you. Let me try.
Let x = the length of the rectangular pond
. (Don't know anyone who would use oblong here.)
Since the area of the pond = length x width = 15,200, the width = 15200 / x
The length of the pond plus the walkway
= x + 4 + 4 = x + 8
(the walkway is on each side so I add 4 twice)
The width of the pond plus the walkway
= 15200/x + 4 + 4 = 15200/x + 8
The area of the pond and the walkway
= 15,200 + 2,104 = 17,304
The area of the pond and the walkway = length of the pond and the walkway times the width of the pond and the walkway so...
17,304 = (x+8
) (15200/x + 8)
That is not the same as what you have below because whoever solved it for you the first time only put a walk on one side of the pond (x+4) instead of both sides (x+4+4 or x+8). Do you understand why the total length is x+8? If not, draw a picture of it.
Now we are going to multiply that out using FOIL (first, outer, inner, last).
17,304 = 15200 + 8x + 8(15200)/x + 64
17,304 = 15264 + 8x + 121600/x
2040 = 8x + 121600/x
Even though the solution you posted has an error in the third line, it is somehow correct at this point. You now multiply both sides by X to get ..
2040x = 8x^2 + 121600
Move all the terms to one side...
0 = 8x^2 - 2040x + 121600
You can solve this several ways. I would just use the quadratic formula if you have been taught that. What your solution below does, is first divide by eight...
0 = x^2 - 255x + 15200
Then factor the problem into
0 = (x-160) (x-95)
So the solutions are x = 160 or x = 95 just like in your solution. Does that help?