I think that this statement is true. I wanted to use set notation to prove this statement. My steps that I took so far are:
1). Show that x is an element in the sets on the left,
x ∈ A ∩ (B ∪ C), x ∈ (B ∪ C)
2). Show that x is an element in the sets on the right.
I used the distributive law to rewrite (A ∩ B) ∪ C to (C ∪ A) ∩ (B ∪ C).
x ∈ (C ∪ A), x ∈ (B ∪ C)
After this, I'm not too sure what step to take next?
It is not true. Take $A=B=\{0\}$ and $C=\{1\}$. Then$$A\cap(B\cup C)=\{0\}\text{ and }(A\cap B)\cup C=\{0,1\}.$$