Proof that (A ∪ B)∘C = A∘C ∪ B∘C where A, B and C are languages How can I prove this identity of languages?

My aproach is the following:

Let A, B and C to be languages, and let x to be an arbitrary string.

(A ∪ B) ⇒ x ∈ A ∨ x ∈ B

How do you proceed?
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.