$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$
beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and
$J = \left\lbrace \sum_{i=0}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z}, a_0 \in 2\mathbb{Z} \right\rbrace$
beeing an ideal of $\mathbb{Z}[x]$ with polynomials with an even constant term. How can I describe the residue class rings of $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J? I have no idea how I should start calculating these. Also I have to find an ideal L in $\mathbb{Z}[x]$, so that $\mathbb{Z}[x]$/L contains 4 elements.
Is polynomial division the way to go here or is it possible to see the residue class rings in the sets I and J?
Define $\phi, \psi$ with your ring as domain and $\mathbf{Z}$ and $\mathbf{Z}/2$ as codomains as folows:
$\phi(f) = f(0)$
$\psi(f)= f(0)\pmod 2$
Verify that both of them are surjective homomorphisms, and apply fundamental theorem about ring homomorphisms