[joe_the_user]

Consistency or inconsistency seems like a red herring here. The proposition concerns conjures independent of ZFC if ZFC is consistent.

Assuming is ZFC is consistent, the theory says there are polynomials whose zeros we could forever fruitless search for but which couldn't be proved to have no zeros in ZFC. We could forever fruitlessly search for a proof of the polynomail having no zero, in fact (putting it this way, returns the situation to something marginally understandable, in fact makes something that would follow from the halting problem).

If ZFC is inconsistent, we actually could prove that this polynomial had a zero and we could prove it didn't have a zero, since we could prove anything, at least anything in the vocabulary of ZFC.