Checking my understanding of right quotient
If $L_1=ab^*c$ and $L_2$ is $b^*c$ then $L_1/L_2$ is $ab^*$.

Note that the $b$'s are not forced to disappear.
For instance (as part of these languages) $\{ab^{10}c\}/\{b^3c\} = \{ab^7\}$.