finite fields - Understanding Primitive Polynomials in GF(2)?

This is an entire field over my head right now, but my research into LFSRs has brought me here.

It's my understanding that a primitive polynomial in $GF(2)$ of degree $n$ indicates which taps will create an LFSR. Such as $x^4+x^3+1$ is primitive in $GF(2)$ and has degree $4$, so a 4 bit LFSR will have taps on bits 4 and 3.

Let's say I didn't have tables telling me what taps work for what sizes of LFSRs. What process can I go through to determine that $x^4+x^3+1$ is primitive and also show that $x^4+x^2+x+1$ is not (I made that equation up off the top of my head from what I understand about LFSRs, I think it's not primitive)?

Several pages online say that you should divide $x^e+1$ (where e is $2^n-1$ and $n$ is the degree of the polynomial) by the polynomial, e.g. for $x^4+x^3+1$, you do $(x^{15}+1)/(x^4+x^3+1)$. I can divide polynomials but I don't know what the result of that division will tell me? Am I looking for something that divides evenly? Does that mean it's not primitive?