Left quotient of a regular language
By definition,
$$
L \diagdown L(A) = \bigcup_{x \in L} \{ w : \delta(q_0,xw) \in F \}.
$$
The idea now is to use the identity
$$
\delta(q_0,xw) = \delta(\delta(q_0,x),w).
$$
You take it from here.