Can ZFC+A and ZFC+negation of A be both inconsistent where A is some conjecture? So I know that a conjecture/statement or negation of it plus ZFC can both turn out to be consistent, which means that a statement is not provable. 
But I would like to go opposite way - and let's say they are both inconsistent. First of all, is this possible, and secondly, if there is an example, what would be a such case?

This cannot happen, assuming ZFC itself is consistent.
If $T$ is any theory then $T+A$ is inconsistent if and only if $T$ proves $\lnot A$. Therefore, if both $T+A$ and $T+\lnot A$ are inconsistent, $T$ must have already been inconsistent.
