"Exactly divisible" means that there will be no remainder if you divide. So let's divide and see what "a" and "b" have to be in order for there to be no remainder:
2x + 4
_______________________________
x^2-1 /2x^3 + 4x^2 + 2ax + b
2x^3 - 2x
---------------------
4x^2 + 2ax + 2x + b
4x^2 - 4
-------------------
2ax + 2x + b + 4
Since we're supposed to have no remainder, the expression at the bottom must work out to be zero. And the expression will be zero if the like terms combine to make zeros. So
2ax + 2x = 0 and b + 4 = 0
2x(a + 1) = 0 b = -4
a = -1