The table above gives selected values of a polynomial function p. Based on the values in the table, which of the following must be a factor of p ?

The correct answer is: (x + 1) (x - 2)

HINT :- if x = a is the root of the polynomial then x - a is the factor of the polynomial 
The polynomial can be written as
(x - a) g(x) where g(x) is another polynomial whose degree is less than the original polynomial.
Given , at x = -1 and x = 2, p(x) = 0 so they are root from table.
x = -1 and x = 2 are roots of the taken polynomial p(x).
The p(x) can be expressed as (x - (-1)) (x - 2) g(x)
p(x) = (x + 1) (x - 2) g(x) : where the degree of g(x) is 2 less then that of p(x)
Now (x + 1) (x - 2) is a factor of the given polynomial