Final answer:
Scientists would stain to determine if there were distinct regions within the organism, specifically a membrane-bound nucleus, which would classify the organism as a eukaryote since prokaryotes lack such structures.
Step-by-step explanation:
To classify a newly discovered unicellular organism as a prokaryote or a eukaryote, scientists would most likely perform a stain to determine if there were distinct regions within the organism, namely a membrane-bound nucleus and other organelles. This is because the presence of a membrane-bound nucleus and membrane-bound organelles like mitochondria is a defining feature of eukaryotic cells. In contrast, prokaryotic cells do not have a nucleus or other membrane-bound organelles.
Both prokaryotic and eukaryotic cells can have a cell wall and cytoplasmic ribosomes, so staining for these would not definitively differentiate between the two. Phospholipids are present in the cell membranes of all cells, so this also would not help determine the type of cell. Since eukaryotes were one of the first types of organisms to evolve, this classification test aligns with the understanding of cellular evolution.