Question: How to simplify $(E \cup F)\cap(F \cup G)$

My process:

$(E \cup F)\cap (F \cup G)$

$((E \cup F)\cap F) \cup ((E \cup F)\cap G)$

$F \cup (E\cap G) \cup (F\cap G)$

And this is where I get stuck. It should simplify to F ∪ EG, but I'm not sure how to turn $E\cap G ∪ F\cap G$ into $E\cap G$.

Answer: 
Hint: Commutation. Then what is $F\cup (F\cap G)~\cup (E\cap G)$?

In fact, do so right from the start. 

$$\begin{align}&~(E\cup F)\cap (F\cup G)\\ =&~ && \text{Commutation}\\=&~F\cup (E\cap G) && \text{Distribution}\end{align}$$

Two east steps; just fill in the middle one and you are done.
Don't you see how to simplify $F\cup FG$? Forget about $E$ for a moment and just draw a Venn diagram for $F$, $G$ and $F\cup FG$.