Let $I=\langle x^3-x-1\rangle$; then you can consider $K=F_3[x]$, which is a field due to irreducibility of $x^3-x-1$. Consider the canonical map $\pi\colon F_3[x]\to K$, and set $\theta=\pi(x)=x+I$.
Now forget that $K$ is actually a quotient ring and identify $a+I$ with $a$ (for $a\in F_3$), which is possible because the restriction of $\pi$ to $F_3$ is injective. Then we have, in $K$, $\theta^3-\theta-1=0$.
Therefore $K$ is an extension field of $F_3$ and $\theta$ is a root of $x^3-x-1\in K[x]$.
What happens to $\theta+a$? Well
due to the field having characteristic $3$; also $a^3=a$ for $a\in F_3$. Hence
Hence $\theta$, $\theta+1$ and $\theta+2$ are distinct roots of $x^3-x-1$ in $K$ and so