Having one supply in the 8X12 rooms would be sufficient. I would install it near one of the exterior walls. The 20X12 room should have two supplies. One supply fed by a 6 inch flexible duct work is not sufficient in my opinion.

Ideally every major branch should have a damper to help balance the air flow. A manual damper is not expensive so I suggest adding them where it makes sense.

If you have the funds replace the flexible duct with hard ducts wherever it is feasible.

mastic seal the flex at both ends, and seal supply boxes to attic floor/ceiling.

support flex with no more than a 1" sag in a 4' section. flex works well when installed properly.

best of luck.

Replace flex duct with hard ducts wherever possible, since hard duct is smoother inside and flows much better for an equivalent duct size.

Also make sure you consider the length of the runs for each of the new ducts. If you have two identical sized runs (e.g., 6" duct), but one has 18 feet of pipe before you get to the vent and the other has 3 feet of pipe before the vent, you will not get identical flow. You might be able to balance it with dampers, but you might also find that you need to upsize the pipe on the longer run so that you get better flow to that room.

Bruce

Yup rigid ducts are an improvement in airflow, especially in straight runs. Easier to hang (support) for long runs as well. Personally I like rigid duct for straight runs and use a combination of rigid elbows and flex for elbows (depending on what I think would work best at the time). Take offs should be correct for what you are planning, there are various types plan accordingly. Dampers would be a really good idea considering you are adding to an existing plan and making adjustments would be crucial. Easy to install, drill two holes and stick the damper in there before you install the duct. Also, requires a section of rigid duct for that damper, keep that in mind.

Sealing your register boots to the wall or floor, as energy rater noted, is also great advice.

Also, most of that quote is labor, the sheet metal, strapping, and sealant/tape you buy at the store really isnt that much. Prepare to buy a few more tools like tin snips and directional aviation snips. And there are a few learning curves to overcome but doable.

If you really want to get into it, you would have to start measuring your trunk branch and check your fan speed and calculate your entire system out to what air goes where. Also factor CFM of the current branch your dealing with and then factor what each room requires and make takeoffs accordingly. So when these CFM go to those rooms they can affect other rooms on that branch.

I have a feeling the contractor wouldnt factor in the exisiting ductwork and how it would affect the overall system, he would just do what you asked.

Rules of thumb for calculations for a quiet system. I go off of this nice PDF concerning HVAC design. It also gives you CFM calcs for flex vs rigid duct.

200 sq in per ton
200 sq in per 400 CFM
400 CFM per ton
600 CFM per 40000 BTU

Enter the rabbit hole.