All right, I've looked at Ars magna, and (if I'm interpreting the rather turgid language correctly) Cardano identifies the conditions under which the method works, and then says something like "I discuss how to handle this when the conditions are not met in the Aliza book". I'm not sure what the Aliza book is, or where it can be found, but I'll keep looking.

He does rather gingerly discuss complex solutions of quadratic equations, but he's even leery of negative numbers, so... (Rather than say the equivalent of "-4 is a solution", he says "4 is a false solution", for instance.)