Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

Question

I'm trying to find the general form of elements in the quotient ring:

$$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$

Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 + 1)$, where $g \in \mathbb{Z}_4 [x]$ and then performing the division algorithm on $g$, however I then realised that $\mathbb{Z}_4$ isn't a field so we don't necessarily have the division algorithm.

Is there a way around this to answer the question?

Thanks