By letting the values, you can check for yourself, and for less marks question you can do it, but if it is asked to prove, then you need to follow proper steps.

Let x ∈ A ∪ (B ∩ C)

x ∈ A or x ∈ (B and C)

x ∈ A or {x ∈ B and x ∈ C}

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ (A or B) and x ∈ (A or C)

x ∈ (A ∪ B) ∩ x ∈ (A ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C)

x ∈ A ∪ (B ∩ C) ⇒  x ∈ (A ∪ B) ∩ (A ∪ C)

Hence, A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) ….. (1)

Again, let x ∈ (A ∪ B) ∩ (A ∪ C)

x ∈ (A or B) and x ∈ (A or C)

{x ∈ A or x ∈ B} and {x ∈ A or x ∈ C}

x ∈ A or {x ∈ B and x ∈ C}

x ∈ A or {x ∈ (B and C)}

x ∈ A ∪ {x ∈ (B ∩ C)}

x ∈ A ∪ (B ∩ C)

x ∈ (A ∪ B) ∩ (A ∪ C) ⇒  x ∈ A ∪ (B ∩ C)

Hence, (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) ….. (2)

From (1) and (2), since both are subsets of each other thus are equal.

i.e., A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)​.