jmc

By Vieta's formulas, $$\begin{array}{rl}(a+4i)+(b+5i)& =10+9i,\\ (a+4i)(b+5i)& =4+46i.\end{array}$$From the first equation, $a+b+9i=10+9i,$ so $a+b=10.$

Expanding the second equation, we get $$(ab-20)+(5a+4b)i=4+46i.$$Hence, $ab=24$ and $5a+4b=46.$

Solving $a+b=10$ and $5a+4b=46,$ we find $a=6$ and $b=4.$ (Note that these values satisfy $ab=24.$) Therefore, $(a,b)=\boxed{(6,4)}.$