The dilation, centered at the origin, with scale factor $-3,$ takes $4 - 5i$ to which complex number?
The dilation, centered at the origin, with scale factor $c,$ takes $z$ to $cz.$

[asy]
unitsize(0.2 cm);

draw((0,0)--(4,-5),dashed);
draw((0,0)--(-3)*(4,-5),dashed);
draw((-15,0)--(15,0));
draw((0,-15)--(0,15));

dot("$4 - 5i$", (4,-5), SE);
dot("$-12 + 15i$", (-3)*(4,-5), NW);
[/asy]

Thus, this dilation takes $4 - 5i$ to $(-3)(4 - 5i) = \boxed{-12 + 15i}.$