"If x - a is a factor of polynomial P(x), then a is a factor of the constant term of the polynomial." - Confused with proof I have recently started learning about polynomials. I've been able to grasp polynomial long division algorithm and the remainder and factor theorems and also a few other common-sense theorems about polynomials. There's just one property of polynomials I don't quite understand the proof of. 
The property:  "If x - a is a factor of polynomial P(x), then a is a factor of the constant term of the polynomial."
There are 2 proofs that I've seen so far that prove this theorem. The first proof I understand and makes complete sense to me. In my view, I think proof 1 is easier to understand.
Proof 1: 
Proof 2:

The second proof is the one I don't understand. More specifically, the part that I don't understand is how:

Can someone please carefully explain how those two expressions are equal to each other? I just don't see how those expressions are equal. I can't find any common factors that have been taken out or what logic has been used to rewrite the expression in that way.

This does not answer your very question, but the theorem seems easy.
If $(x-a)$ divides $P(x)$ then $P(a)=0$. But $P(a)=a(p_na^{n-1}+\cdots+a p_2+p_1)+p_0$ and $p_0=-a(\cdots)$.
