You need to use the division algorithm here; you can use it since $x^2 + 1$ is monic in the ring $\mathbb{Z}_4[x]$. For every polynomial $f \in \mathbb{Z}_4[x]$, by the division algorithm we can write it as
$f = (x^2 + 1)q(x) + r(x)$
where the degree of $r$ is bigger than or equal to zero, less than 2. You can now see that the cosets in the quotient are of the form
$(\text{linear polynomial}) + I$
where $I$ is the ideal generated by $x^2 + 1$. Now the linear polynomial can be written as $ax + b$ for $a,b \in \Bbb{Z}_4$. But then recall that $x^2 + 1 = 0$ in the quotient, so that we get ring a new ring (the quotient ring) where multiplication between cosets $A + I$ and $B + I$ is defined by $(A + I)(B+ I)= (AB) + I$ and where we have the relation $x^2 + 1 = 0$.