Find all irreducible polynomials of degree ≤4 in Z₂[x].
Hint: If f(x) is a polynomial in your ring, then f has a linear factor x-a if and only if f(a) = 0 in the ring. Since your ring is small, you can easily check which polynomials have linear factors.
For quadratic factors, just write it out as (x^2 + bx + c)(x^2 + sx + t) = f(x) (for degree 4 polynomials) and expand and figure out what you get. There are only 16 monic polynomials of degree 4 in your ring, 8 of degree 3, and 4 of degree 2, so it shouldn't be too hard to figure out which ones are irreducible.