Let's say we have $\mathbb{Z}_3[X]\mathbin{/}(7x^2 + 1)$, such that we take the quotient of the polynomial ring $\mathbb{Z}_3[X]$ with that specific ideal.
Is this still a ring, and does it make sense to ask what the result of $(x^2 + 3x^3) \cdot (13 + 7x)$ is in it? What's the procedure for it?
On
Is it still a ring?
Yes, for an ideal $I\subseteq R$ we have that $R/I$ is the quotient ring.
Does it make sense to ask what the result of $(x^2 + 3x^3) \cdot (13 + 7x)$ is?
Yes. In this ring $7x^2+1=0$ and $3=0$, so $x^2=-1$. This means that every polynomial is on the form $$a+bx$$ where $a,b\in\Bbb Z_3$. Since any polynomial may be reduced to a polynomial of lower degree with $x^2=-1$.
Example: $$\begin{align}(x^2 + 3x^3)(13 + 7x)&=(-1-3x)(1+x)\\&=-1-4x+3x^2\\&=-1-4x-3\\&=-4-4x\\&=2+2x\end{align}$$
In fact, $$\Bbb Z_3[x]/(7x^2+1)\cong\Bbb Z_3[i]$$
In the same way that the quotient ring $\mathbb{Z}_3 = \mathbb{Z}/(3)$ consists of cosets of the ideal $(3) = 3\mathbb{Z} \subset \mathbb{Z}$, i.e., equivalence classes under the congruence modulo $3$ equivalence relation on $\mathbb{Z}$: $$ m + (3) = n + (3) \quad\iff\quad m - n \in (3) \quad\iff\quad m \equiv n \pmod{3}, $$ the polynomial quotient ring consists of cosets of the ideal $(7X^2 + 1) \subset \mathbb{Z}_3[X]$, i.e., equivalence classes under the congruence modulo $(7X^2 + 1)$ equivalence relation on $\mathbb{Z}_3[X]$: \begin{align} p + (7X^2 + 1) = q + (7X^2 + 1) \quad&\iff\quad p - q \in (7X^2 + 1) \\ \quad&\iff\quad (7X^2 + 1) \mid (p - q) \\ \quad&\iff\quad p \equiv q \pmod{7X^2 + 1}. \end{align}
It is common to abuse notation and suppress the notation for the ideal/equivalence class in favor of just writing the representative of the class, with the understanding that all expressions are considered modulo the ideal. What's important is to keep in mind that in your quotient ring, $7X^2 + 1 = 0$, or more simply since $7 \equiv 1 \pmod{3}$, the relation is $X^2 + 1 = 0$, so for example, $$ X^3 = X(X^2) = X(-1) = -X. $$
And indeed, since the polynomial $X^2 + 1$ is of degree $2$, every element of the ring is equivalent to a class represented by a linear polynomial $aX + b$ for some $a, b \in \mathbb{Z}_3$. In other words, this is a finite ring with only $3^2 = 9$ elements. If you wanted, you could even write down addition and multiplication tables for it!