Show L1 /L2 is regular
Let $A$ be a DFA accepting $L_1$, and let $Q$ be its set of states. Denote by $L_A(q)$ the set of words which cause $A$ to move from its initial state to state $q$. For each state $q$ and word $w$, one of the following holds (exercise):


For every $x \in L_A(q)$, $xw \in L_1$.
For every $x \in L_A(q)$, $xw \notin L_1$.


We now modify $A$ by determining its accepting states as follows. A state $q$ is accepting if for some word $w \in L_2$, $xw \in L_1$ for all $x \in L_A(q)$. It is not too hard to show that the modified automaton accepts $L_1/L_2$ (exercise).