jmc

Setting $x=i-3,$ we get $$a(i-3{)}^3+9(i-3{)}^2+a(i-3)-30=0.$$Expanding, we get $42-21a-54i+27ai=0,$ so $a=2.$

Then the polynomial is $2x^3+9x^2+2x-30.$ Since $i-3$ is a root, $-i-3$ is also a root, which means $$(x-i+3)(x+i+3)=x^2+6x+10$$is a factor. We can then say that the polynomial factors as $(2x-3)(x^2+6x+10).$ Thus, $b=\frac{3}{2},$ and $$a+b=\frac{3}{2}+2=\boxed{\frac{7}{2}}.$$