Does the distributive property apply to regular expressions? Let $(Q,\Sigma,\delta, q_0, F)$ be a finite automaton, does 
$$
(q_1a+q_2b)c=q_1ac+q_2bc
$$

hold for any $q_1,q_2\in Q$ and $a,b,c\in \Sigma$?

Here $Sa=\{\delta(q, a)\mid q\in S\}$ and $qa=\{q\}a$ for any $S\subseteq Q$, $q\in Q$ and $a\in\Sigma$. The operation "$+$" means set union.
since someone has edited the question, the question seems much more meaningful , and yes the above equality holds , here the equations 

taking LHS of the equation 

= ( $q_1$a  $+$ $q_2$b ) c

=$\delta$ ( (   $q_1$a  $+$ $q_2$b ) , c )  { as you defined Sa={ δ (q,a) ∣q ∈ S} }

=$\delta$ ( (   $q_1$a )  $\bigcup$   $q_2$b ) , c ) 

=$\delta$ ( ( $q_1$a),c ))    $\bigcup$   $\delta$ ( (    $q_2$b ) , c )  

=$q_1$ ac $+$ $q_2$bc  

which is equal to RHS.