Show that the language of strings not in the union of two regular languages is regular
There are several ways to show that a language is regular (check the question How to prove a language is regular?)

Specifically for the language in your question, start with DFAs for $L_1$ and $L_2$ and try to construct an NFA for $X(L_1,L_2)$ using them. More details below:


   Note that $X(L_1,L_2) = \overline{L_1} \cap \overline{L_2}$.
 From the DFA of $L_1$ construct the DFA of $\overline {L_1}$ (making any final state not-final, and vice-versa). Do the same for $L_2$.  Intersection of regular languages can be constructed via product machine (see this question).
 [Of course, if you already know that a complement of a regular language is also regular, and so the intersection of two regular languages – you are done without constructing those DFAs..]