jmc

Let $f(x)$ be the polynomial of degree $13$ and let $q(x)$ be the quotient when $f(x)$ is divided by $d(x)$. Let $r(x)=3x^3+4x^2-x+12$. Then we have $$f(x)=d(x)\cdot q(x)+r(x).$$where $\mathrm{deg}q=7$.

Since $\mathrm{deg}r=3$, we need to have $\mathrm{deg}(d\cdot q)=\mathrm{deg}f$ which means $\mathrm{deg}d+\mathrm{deg}q=\mathrm{deg}f$. So $\mathrm{deg}d=13-7=\boxed{6}.$