The reason you're having trouble seeing the second step is because it's not true. If you multiply out you will have a polynomial of degree 5 in which is not equal to the degree 3 polynomial
Does this clear things up? Let me know if there is a point of confusion.
Also, not sure if you have seen this or not, but it's worth checking out the "Freshmen's Dream" theorem Freshman's dream - Wikipedia, the free encyclopedia. Since 3 is a prime this theorem tells us that IN , which was what you concluded in your post above.