Let $F_5$ be the field of integers modulo $5$.
I am trying to find out if $x^2 + x + 2$ is primitive or not.
So first we see that the divisors of $24$ are $1,2,3,4,6,8,12$.
We find $$x^2=-x-2$$ $$x^3=-x^2-2x$$x $$x^4 =(-x-2)^2=x^2+4x+4=-2x+2$$
How does $x^2+4x+4=-2x+2$?
How does $x^6=x^2-4x+4=x^2+x-1$?
How does $x^8=-x^2+2x-1=-2x+1$?
From this how can we deduce that the order of $\alpha$ is not $1,2,3,4,6,8$?Is it because they dont equal $1$?
If $x^6=2$ how does $x^{12}=-1$
We can see that $x^{24}=1$ so the polynomial is in fact primitive.
The polynomial $2+X+X^2$ is the "default primitive polynomial" for $\mathbb{F}_{5^2}$, see here. Regarding your last question: If $X^6=2$ then $X^{12}=2^2=-1$ over a field of characteristic $5$. We have $$ X^3=-X^2-2X=-X+2, $$ and $$ X^4=X^2+4X+4=(-X-2)+4X+4=3X+2=-2X+2. $$