Your ex? Why do your care about your ex's car? Haha.
It shouldnt be burning that much oil. Their is definitley an issue, but needing a new engine is not needed. The issue may require a tear down. Its possible a piston ring cracked or is somehow allowing oil into the combustion chamber. Either valve guides, piston rings, cracked block, or all of the above.
For having 150,000 miles, it should not have this problem. Its not common for a 2005 to burn oil. A 20+ year old car its acceptable to some extent, but its not common even on the E30's. Get a quote from a couple different shops on an engine rebuilt. The reason their expensive is because of the labour involved. It can take 15~ hours to tear one down, fix it, and put it back together. Its very possible to find an alternate engine if need by. Their are many salvage and junk yards that may carry one. A used engine for a 2005, they will probably still be in the $6k range.
If this isnt something cost effective for you, then unfortunaltey you may be best to sell the car and replace it with a different one. Any other questions feel free to ask.
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