Is 2 a root of the polynomial? x^4 + 2x^3 - 9x^2 - 2x + 8

Yes.

first factor, it’ll come out to:
(x+1)(x-1)(x-2)(x+4)

then just solve each individual factor, but the only one you’re interested in is (x-2) so:

x-2=0
x=2

2 is a root of that polynomial
(the other roots are -1, 1, and -4)

The correct answer is 7