Is a pumped-up version of endocytosis where large macromolecules, or even entire microscopic organisms, are engulfed and "eaten" by another cell.

The answer is PHAGOCYTOSIS.
Phagocytosis refers to the process by which a cell engulf a material or another cell. In unicellular organisms, phagocytosis is a mode of feeding while in eukaryotes, phagocytosis is the mean by which the defensive agents of the immune system attack foreign bodies which are found in the system. 

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Ithink the answer is nuclear membrane