2004 BMW 330xi going through oil

I have a 2004 BMW 330xi with 115,000k. Over the past few years I have to put oil in the car frequently. Four years ago it was every 1500-2000 miles, but It has now gotten to the point where it needs a quart of oil every 400 miles. There are no leaks under the car. There is no smoke coming out of the back. The BMW dealer has no clue. They’ve replaced some gaskets and a few other things but nothing helps. Another mechanic recommended changing the oil from the regular synthetic to a blend and driving the car at higher rpm’s. That didn’t work either.

Where is all this oil going? I am worried that something is really wrong and I’m going to loose the engine.
If anyone has any clue I’d be really grateful for some advice. Hoping I don’t have to get rid of this car. It’s in great shape

It may be the smoke is only coming out at first, after starting the engine, so you don’t notice it. Is the tail pipe black and greasy? that could be a symptom of worn valve guide seals, a relatively easy problem to fix. Well, maybe “easy” isn’t he right term. But it doesn’t require the engine or transmission to be removed usually.