Final answer:
An organism with a cell that has a nucleus and other membrane-bound organelles is a eukaryote, indicating it belongs to the Domain Eukarya.
Step-by-step explanation:
If a living organism contains a cell with structures such as organelles, a nucleus, mitochondria, the endoplasmic reticulum (ER), and a complex nuclear membrane, we can conclude that the organism is a eukaryote. This is because eukaryotic cells are characterized by having these membrane-bound organelles and a nucleus, which are absent in prokaryotic cells. In contrast, prokaryotic cells, such as those belonging to the domains Bacteria and Archaea, lack a nucleus and membrane-bound organelles and often have a single circular chromosome. Therefore, the presence of organelles within the cell indicates that the organism belongs to the Domain Eukarya and is a eukaryote, which answers the student's question.