Does the distributive property apply to regular expressions?
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.