Upon investigation, it was noticed that there was indeed a correlation between the difference between the squares of values above and below a number and the number itself.
Soon arriving with this equation for part 1.
(x+1)² - (x-1)²
=x² + 2x + 1 - x² + 2x - 1
=4x
Therefore the difference between the squares of (x+1) and (x-1) is 4x.
When;
x=-1, 0²-(-1)²=-1, -1x4=-4, False
x=0, 1²-(-1)²=0, 0x4=0, True
x=1, 2²-0²=4, 1x4=4, True
x=2, 3²-1²=8, 2x4=8, True
x=3, 4²-2²=12, 3x4=12, True
x=4, 5²-3²=16, 4x4=16, True
x=5, 6²-4²=20, 5x4=20, True
x=10, 11²-9²=40, 10x4=40, True
x=50, 51²-49²=200, 50x4=200, True
x=539, 540²-538²=2156, 539x4=2156, True
etc...
Based off such evidence we can say that the difference of two squares above and below a positive integer(including 0) is 4 times the said integer(x).
With the extension(part2) in the same light...
(x+2)²-(x-2)²
= x²+4x+4-x²+4x-4
=8x
Therefore the difference between the squares of (x+2) and (x-2) is 8x.
When;
x=0, 2²-(-2)²=0, 0x4=0, True
x=1, 3²-(-1)²=8, 1x8=8, True
x=2, 4²-0²=16, 2x8=16, True
x=3, 5²-1²=24, 3x8=24, True
x=4, 6²-2²=32, 4x8=32, True
x=5, 7²-3²=40, 5x8=40, True
x=200, 202²-198²=1600, 200x8=1600, True
etc...
Based off such evidence we can say that the difference of two squares 2 above and 2 below a positive integer(including 0) is 8 times the said integer(x).