Splitting over some extension of $\mathbb Z_3$ I want to write $x^3 + 2x + 1$ as a product of linear polynomials over some extension field of $\mathbb Z_3$.
I know that the roots will lie in a field that will be isomorphic to the field $\mathbb Z_3[x]/<x^3 + 2x + 1>$ but not sure how to proceed further. 

As your polynomial is irreducible, the Frobenius automorphism $a \mapsto a^3$ gives  $$T^3+2T+1 = (T-\alpha)(T-\alpha^3)(T-\alpha^{3^2})$$ where $\alpha = x + (x^3+2x+1)$ is one of its roots in $\mathbb{Z}_3[x]/(x^3+2x+1)$. 
Then you compute $\alpha^3 = x^3+(x^3+2x+1)= x+2 +(x^3+2x+1), \alpha^{3^2} = (x+2)^3 + (x^3+2x+1) = x^3+2^3 +(x^3+2x+1) = x+1+(x^3+2x+1) $ and hence 
$\qquad\qquad T^3+2T+1 = (T-x)(T-x+1)(T-x+2)$ in $\mathbb{Z}_3[x]/(x^3+2x+1)$.
