The history of a certain rock began as a solidified melt, which was then was broken apart by wind, water, and ice. the broken pieces came to rest in a low basin near a coastline and then were buried. there these pieces became heated and squeezed such that the minerals in these pieces altered into other minerals. what type of rock is this?

The answer is metamorphic rock. 

Based on the explanation it explains the different changes and transformations that it has undergone. The solidified rock formed from melted rocks is in a state of being an igneous rock. The weathering of this rock going to a low basin and coastline can mean that it is possible that it turned into a sedimentary rock based on its situation. Finally, the heat and pressure in the coastline made sedimentary rocks turn into another type of rock which is the metamorphic rock.