Proof that (A ∪ B)∘C = A∘C ∪ B∘C where A, B and C are languages
Just keep going. You've written what it means for $x$ to be in $A\cup B$. Now write out what it means for a string $y$ to be in $(A\cup B)\circ C$.  Do the same kind of thing on the right-hand side and relate the two statements you get.