Answer
It is not a factor of $f(x).$
Work Step by Step
The Remainder Theorem says that the remainder when a function $f(x)$ is divided by $(x-r)$ is $f(r)$.
Hence $f\left(-\frac{1}{3}\right)=3\cdot\left(-\frac{1}{3}\right)^4+\left(-\frac{1}{3}\right)^3+1+1=2\ne0$, therefore $\left(x+\frac{1}{3}\right)$ is not a factor of $f(x)$ by the factor theorem.
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and vice versa.