How to find all irreducible polynomials in Z2 with degree 5? I am totally lost on how to do this one. I am supposed to accomplish the following:
Find all irreducible polynomials in $\mathbb{Z}_2[x]$ with degree $5$. 
I may use the fact that x, $x+1$ and $x^2+x+1$ are the irreducibles of degree less than or equal to 2
If someone could provide a step by step explanation of how to do this, that would be amazing! Thanks in advance!
P.S.: This is not a homework, but is a question on a past exam that I couldn't answer and I want to know how to do it. 

The following argument is specific to the question asked. Although I am
by no means a trained professional, please do not attempt this at home with
polynomials of larger degree.
The irreducible polynomial must be of the form
$x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + 1$, that is, the degree-$5$ term
as well as the constant term must be nonzero. Of the $16$ possible
choices of $(a_4,a_3,a_2,a_1)$ we can eliminate the $8$ vectors of
even Hamming weight because the corresponding degree-$5$ polynomials 
have $1$ as a root. So, we are down to $8$ polynomials of which 
$6$ are necessarily
irreducible because the $30$ elements of $\mathbb F_{32}-\mathbb F_2$
all have order $31$ and come in sets of $5$ conjugates. Starting with
the lexicographically first polynomial $x^5 + x + 1$, we readily
discover via long division that
$$x^5 + x + 1 = (x^2+x+1)(x^3+x^2+1)$$ 
and so $x^5 + x + 1$ and its reciprocal $x^5+x^4+1 = (x^2+x+1)(x^3+x+1)$ 
are reducible polynomials. Thus, the irreducible polynomials of 
degree $5$ are
$$\begin{align}
x^5+x^2+1 &\quad x^5+x^3+1\\x^5+x^3+x^2+x+1 &\quad x^5+x^4+x^3+x^2+1\\
x^5+x^4+x^2+x+1 &\quad x^5+x^4+x^3 + x + 1
\end{align}$$
