Answer
Remainder = $2$
$(x+\frac{1}{3})$ is not a factor of $f(x)$.
Work Step by Step
The Remainder Theorem states that when a function $f(x)$ is divided by $(x-R)$ , then the remainder will be: $f(R)$.
We have: $f(x)=3x^4+x^3-3x+1$
$f(\frac{-1}{3})=(3)(\frac{-1}{3})^4+(\frac{-1}{3})^3-(3)(\frac{-1}{3})^3+1=2 \ne 0$
The Factor Theorem states that if $f(a)=0$, then $(x-a)$ is a factor of $f(x)$ and vice versa.
Therefore, by the Factor Theorem $(x+\frac{1}{3})$ is not a factor of $f(x)$.