First differences are ... 0-7 = -7, -1-0 = -1, 4-(-1) = 5, 15-4 = 11, 32-15 = 17.
Second differences are -1-(-7) = 6, 5-(-1) = 6, 11-5 = 6, 17-11 = 6.
The second differences are constant at 6, so the table values would be best fit by a function that is ...
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The degree of the polynomial function corresponds to the level of differences that are constant (when the x-values are evenly spaced).
The quadratic function that matches these values is
... y = 3x² +2x -1
... y = (3x-1)(x+1) . . . . in factored form
