Consider the polynomials $2x+3$ and $3x^2+5 \in \mathbb{Z}[x]$. Does there exist $q(x)$ and $r(x)$ such that $3x^2+5 = (2x+3)q(x)+r(x)$?

Question

Since degree of $3x^{2}+5$ is $2$ and that of $2x+3$ is $1$, if possible, let there exist $q(x)=ax+b$ and $r(x)=c$, where $a,b,c \in \mathbb{Z}$ such that $3x^2+5= (2x+3)q(x)+r(x)$. This implies $(3x^2+5)= (2x+3)(ax+b)+c \implies a= \frac{3}{2}, b= -\frac{9}{4}, c= \frac{47}{4}$, none of which belong to $\mathbb{Z}$. So there does not exist any such $q(x)$ and $r(x)$. Is my reason correct?