I have a feeling that I've seen this question before, but here goes:

1. Not enough. Pick -3 and -5: they're consecutive odd integers and at least one of them is negative (actually, both are), with sum = -8. But pick -101 and -103 and, while they still conform to the constraint, their sum is -204.

2. Not enough either for the same reason: pick 3 and 5 or 101 and 103.

But put both stmts together and you've got smth: when taking into consideration both restrictions at the same time, you realize that you're looking for a pair of consecutive odd numbers, with one of them positive and one negative. There's only one such pair: -1 and 1. You really don't have to look far for it either, since that "one positive, one negative" restriction means that you're looking around zero.
The sum of the pair will be 1 - 1 = 0. C it is.