Difference between $\mathbb{Q}(x)$ and $\mathbb{Q}[x]$? In general I'm having a hard time understanding what exactly $\mathbb{Q}(x)$ is.... i.e. () circular brackets.

This notation is used when $Q \subseteq K$, where $K$ is some field and $x \in K$. Then $Q[x] = \{f(x) : f  \text{ is a polynomial over } Q \}$, while $Q(x) = \{f(x) / g(x) : f,g  \text{ are a polynomials over } Q \text{ and } g(x) \neq 0 \}$. If $x$ is algebraic over $Q$ then they are the same i.e. $Q[x] = Q(x)$. That's why you may be confused...
