My try:
Suppose $x \in A \cup (B \cap C)$. We know that $x \in A$ or $x \in B \cap C$, by the definition of Union. So we have two cases:
Case one: Suppose $x \in A$. We can say that both $x \in A \cup B$ and $x \in A \cup C$. By the definition of intersection, we know that $x \in (A \cup B) \cap (A \cup C)$.
Case two: Suppose $x \in B \cap C$. By the definition of intersection, we know that $x \in B$ and $x \in C$. We can say that $x \in A \cup B$ and $x \in A \cup C$. Hence, we know that $x \in (A \cup B) \cap (A \cup C)$.
So we have covered both possibilities and arrived at the conclusion that $x \in (A \cup B) \cap (A \cup C)$. Hence, by the definition of subset, we know that $A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C)$.