I have two polygons.
The first one has a white background and brown boundary. To other has the pink background.
In QGIS, how is it possible to reshape to first polygon to the boundary of the other polygon?
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PostGIS has ST_Snap, which could do most of what you need. I'm not current enough with QGIS though, so perhaps you can't use that through Processing, but would have to manually convert it to a database first and then load it through the DB manager.
However, do note that even ST_Snap is not completely deterministic, as it uses a heuristic, so sometimes the results won't be what you wanted.