Trying to convert garage into family room. I want to run ductwork from
basement to attached garage.

HVAC guy wants to put a vent under my front window. Problem is that
the end of the garage (where the vent will go), is about 4' past the
end of the basement. So if I took the ductwork out of the last
basement bay, it would be about 4', and a few garage bays, away from
the target vent location.

So I need the ductwork to come out of the basement wall, hang a left,
go "through" a joist or two, and enter another bay. Then it can run
down that bay and end in the vent.

So how should I adjust my floor framing to accomodate this duct? Is it
possible to end my joist 6" before the wall, and "box it out". Will
this have negative impacts on the strength of the floor?

Thanks,

Chris