Protein synthesis occurs in the:

Protein synthesis occurs primarily in the ribosomes, which are the cellular structures responsible for translating messenger RNA (mRNA) into polypeptide chains, ultimately forming proteins. Ribosomes can be found either floating freely within the cytoplasm or attached to the endoplasmic reticulum, which collectively creates what is known as the rough endoplasmic reticulum (RER). The process of translation involves the ribosome reading the sequence of the mRNA, with transfer RNA (tRNA) bringing the corresponding amino acids to build the protein according to the genetic instructions.

In contrast, while the nucleus is the site of DNA transcription where mRNA is synthesized, it is not where protein synthesis takes place. The mitochondria are primarily involved in energy production through oxidative phosphorylation, rather than in protein synthesis. The Golgi apparatus plays a crucial role in modifying, sorting, and packaging proteins for secretion or delivery to other organelles but is not where the actual synthesis occurs. Therefore, the ribosomes are the key sites for the synthesis of proteins, making this the correct choice.