Find a polynomial of degree 5 that is irreducible over $\mathbb{Z}_3$. Can someone please let me know if this looks ok? Thanks in advance!
An irreducible degree $5$ polynomial over $\mathbb{Z}_3$is one such that $f(0)\neq0,f(1)\neq0,f(2)\neq0$.
Take e.g. $p(x)=x^5+x^4+x^3+x^2+1$ 
$p(0)=1$
$p(1)=2$
$p(2)=1$
And,  $$x^2,x^2+1, x^2+x+1\ \nmid\ x^5+x^4+x^3+x^2+1.$$
$\therefore x^5+x^4+x^3+x^2+1$ is an irreducible polynomial over $\mathbb{Z}_3$.

Your problem is to find an irreducible polynomial of degree $5$ over $\mathbb Z_3$. You haven't found one yet, because
$$x^5+x^4+x^3+x^2+1=(x^2+x+2)(x^3+2x+2).$$
Note that the irreducible (monic) polynomials of degree $2$ are $x^2+1$ and $x^2+x+2$ and $x^2+2x+2$. You need to check all three of them as possible factors.
Hint. Try $x^5+2x+1$.
