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The organelle primarily responsible for synthesizing a large amount of protein for export in a cell is the endoplasmic reticulum. Specifically, the rough endoplasmic reticulum (RER) is studded with ribosomes, which are the sites of protein synthesis. The presence of ribosomes on the RER allows it to efficiently produce proteins that are either secreted from the cell or incorporated into the cell membrane.

While ribosomes play a crucial role in protein synthesis, they do not function as standalone organelles capable of exporting proteins. Instead, they are the molecular machines that translate messenger RNA (mRNA) into polypeptide chains. The rough endoplasmic reticulum processes these newly formed proteins and directs them into vesicles for transport to the Golgi apparatus, where they are further modified and packaged for export.

Mitochondria are primarily involved in energy production through ATP synthesis and do not play a direct role in protein export. The nucleus contains the cell's genetic material and is responsible for transcription but is not directly involved in protein synthesis or export either. Therefore, the endoplasmic reticulum is the correct answer as it serves as the primary site for synthesizing proteins meant for export.