When the x values are evenly spaced, as they are here, the second differences are constant when the function is quadratic.
1st table:
.. 2 -4 = -2; 1 -2 = -1; 0.5 -1 = -0.5 . . . . first differences
.. -1 -(-2) = 1; -0.5 -(-1) = 0.5 . . . . . . . . . second differences are different
2nd table:
.. 128 -135 = -7; 105 -128 = -23; 72 -105 = -33 . . . . first differences
.. -23 -(-7) = -16; -33 -(-23) = -10 . . . . . . . . . . . . . . . second differences are different
3rd table:
.. -23.2 -(-23.4) = 0.2; -23 -(-23.2) = 0.2 . . . . . first differences are constant. This is a linear function.
4th table:
.. 56 -90 = -34; 26 -56 = -30; 0 -26 = -26; -22 -0 = -22; -40 -(-22) = -18
.. -30 -(-34) = 4; -26 -(-30) = 4; -22 -(-26) = 4; -18 -(-22) = 4 . . . . second differences are constant at 4.
The 4th selection is appropriate.
