jmc

Suppose our polynomial is equal to$$a{x}^3+b{x}^2+cx+d$$Then we are given that$$-9=b+d-a-c.$$If we let $-a=a^{\prime }-9,-c=c^{\prime }-9$ then we have$$9=a^{\prime }+c^{\prime }+b+d.$$This way all four variables are within 0 and 9. The number of solutions to this equation is simply $\left(\genfrac{}{}{0ex}{}{12}{3}\right)=\boxed{220}$ by stars and bars.