A new floor can only be installed over ONE existing floor. The terrazzo is floor #1, the engineered is floor #2.

Just rip up the wood and work with your new floor. If you put down a new floor over these other two, you will lose all warranty and you will cause issues with DOOR CLEARANCE. Yah...check to see how much room you have between the bottom of the door and the engineered hardwood. And now look at the transition you will create in the doorway. The floor height variation will be big.

You have water damage on the wood. If you put something over top you will create an anaerobic condition that will allow bacteria and mold grow into a lovely germ soup. We've seen it before. And if you put down a floor on top of water damaged wood over a terrazzo floor you will see it and smell it too.

Ask yourself how much you look forward to REDOING this project in 1-2 years?

@SJ McCarthy one of the issues in my house is that there are layers of tile in other areas, so the majority of the house is an inch higher than this room. Im currently getting new doors, so doors and room transitions aren't an issue, in fact, id like to raise this floor up at least a half inch to an inch to be closer to the hallway. The water damage happened a few years ago when I had someone cleaning my windows every couple of weeks and they were hosing the window-the area has been dry for a couple of years, but there is an area of about 6 inches deep that needs to be either taken out or sanded down to be smooth (or maybe there is an underlayment I can use-not sure). I realize that the ideal way to do this would be to remove all of the old floor and lay new plywood on top of the terrazzo, and then add the new floor to make it level with the rest of the house, but that is out of my budget. SO... i'm wondering if I can use this old floor which is plywood and veneer, as a base for the new floor. Could I float a floor or nail it down to this floor?

How would mold grow under there if the floor has been dry for a couple of years?

Rip 'em all out and refinish the terrazzo please. It's the best floor of all.

It’s not nice terrazzo and it would be about an inch and a half below the hallway.

So here is what you are doing: you are slowing reducing your ceiling height (which is a selling point = higher ceilings = higher value in the home) with each layer of flooring.

Just because SOMEONE else has made poor flooring decisions (made because they didn't have the budget to do it properly) doesn't mean you need to follow suit.

Each layer you put down is another layer you will HAVE to pull out. It has to come out SOMETIME!!!! You can purchase new doors and new trim and new counter tops and new outside doors and new siding...etc...

OR you can do the 'right thing' and get rid of the HORRIBLE flooring set up. You will pay for it today...or you will pay DOUBLE tomorrow.

If you INSIST on doing this...then use the LOWEST priced flooring on the market = $0.79sf laminate and float the darn thing! Don't put down ANYTHING of value....NOTHING! Don't throw a 'medium' amount of money at this situation so that you can save a tiny amount today.

You WILL PAY MORE. And remember: anything 'man made' (vinyl or bamboo) will NOT BE available 12 months from now. Which means you will NEVER match what is in that room.

And that means it is a WASTE OF MONEY!

If you must waste money (and you will be) you really should waste as LITTLE as possible. Find a nasty laminate and throw it down. You will rip it out in 7-10 years anyway.

And no....don't trim your doors. It is the WRONG PLACE to put your money. Again = TOTAL WASTE OF MONEY!

It's your call. How much money do you WANT to waste? That's what you need to ask yourself. IF you say, "Waste money? None! I never want to waste it! I can't afford it!" then you need to leave the flooring situation alone and start REMOVING the cr@ppy floors you have in the house and start saving for a REAL solution.

@SJ McCarthy Im not ripping all of my floors out, maybe one day, but I cant now. My kitchen is on this floor, my doorways are ALL leveled with this floor! Even if I ripped it all out, id have to bring it up to make all of these doorways normal, so I need that ONE floor to be taller.

OK. Your money. Your house.

I advise you spend VERY little to do it. That means a floating laminate.

Anything more and you are flushing money down the toilet.

Good luck. I hope you get more than 7years out of it.