what does this notation mean? (Polynomials as vector space)

Question

Let V$=\mathbb{F}[x]$, determine if the quotient space V/W is finite dimensional when W$=x^nV$. Is $W$ the space when multiplying every polynomial by $x^n$?

Also is the answer to that question finite dimensional because I can construct a basis $\{V,x+V,...,x^{n-1}+V\}$ for $V/W$?

Answer 1

Yes, $x^n V$ means $\{x^n v: v \in V\}$. And yes, that is a basis.

Answer 2

Yes, your understanding is correct.