$F\{x,y\}/(xy-yx-x)$ is primitive

Question

Let $F$ be a field of characteristic $0$ and $F\{x,y\}$ denote the free algebra over $F$ generated by $x,y$. Then show that $R=F\{x,y\}/(xy-yx-x)$ is primitive. I tried to use the usual trick that to embed $R$ inside the endomorphism ring of an abelian group $M$ and show $M$ is irreducible as $R$ module. But I did not succeed. For example the subring of $\text{End}_FF[t]$ generated by $t$ and $-t\frac{d}{dt}$ satisfies the given commtation relation but $F[t]$ is not irreducible over this ring .