The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$ where $m \in \mathbb{N}$

Classic examples of how one can treat such rings is to find relationships like $$\mathbb{Z}[x]/(1-x,p) \cong \mathbb{Z_{p}}$$ for prime $p$.

However, I often struggle to intuitively understand what the elements of such rings are, and they to compute using them.

For example, what do the elements of the set $\mathbb{Z}_{2}[X] / (x^4+1)$ actually look like. Can this set be described in a clearer way to help understand the way the rings work.

Is there some general way of describing elements of such rings so that isomorphisms and computations become easier to handle?

Apologies if this question is not sufficiently clear.