Final answer:
Proteins synthesized on free ribosomes remain in the cytosol or are directed to specific organelles and do not pass through the Golgi apparatus. Attached ribosomes on the rough ER synthesize proteins that are involved with the endomembrane system. The correct intracellular path for proteins that stay within the cell is from the nucleus to free ribosomes and then to the cytoplasm.
Step-by-step explanation:
Protein Synthesis and Intracellular Transport
When discussing the path that a protein takes inside a cell if it is going to stay within the cell, we can look at the process from the point where the protein is synthesized to where it ultimately resides. Proteins synthesized on free ribosomes in the cytoplasm stay in the cytosol or are targeted to certain organelles such as the nucleus, mitochondria, or chloroplasts. They do not typically travel through the Golgi apparatus.
In contrast, proteins synthesized on attached ribosomes, which are bound to the endoplasmic reticulum, usually become part of the endomembrane system. In part (b) of the referenced information, we see that these ribosomes on the rough ER initiate the protein synthesis, which then can be processed by the endomembrane system, possibly involving the Golgi apparatus if the protein is to be secreted, or become a part of the plasma membrane or other organelles. Hence, a protein that remains within the cytoplasm may bypass the Golgi apparatus entirely.
Therefore, among the given options, A: nucleus-- free ribosomes --cytoplasm seems to represent the correct pathway for the synthesis and staying of a protein within the cytoplasm without interacting with the endomembrane system.