Closure of context-free languages under regular quotient
Let me prove the following more general statement:


  Every full trio is closed under right quotient by a regular language.


Let $\mathcal{L}$ be the full trio, let $L \in \mathcal{L}$ be a language over $\Sigma$, and let $R$ be a regular language over $\Sigma$. Let $\Sigma' = \{ \sigma' : \sigma \in \Sigma \}$. Define the homomorphism $h\colon \Sigma \cup \Sigma' \to \Sigma$ by $h(\sigma) = h(\sigma') = \sigma$, and define the homomorphism $k\colon \Sigma \cup \Sigma' \to \Sigma$ by $k(\sigma') = \sigma$, $k(\sigma) = \epsilon$. Then
$$
L/R = k(h^{-1}(L) \cap \Sigma^{\prime *} R).
$$