The part of the cell that has an extensive network of membranes with ribosomes on its surface is the rough endoplasmic reticulum (RER).

Here's why:

* Extensive Network of Membranes: The RER is a complex, interconnected system of flattened sacs (cisternae) and tubules.

* Ribosomes: Ribosomes, the cellular organelles responsible for protein synthesis, are attached to the surface of the RER.

* Function: The RER plays a crucial role in protein synthesis and modification. The ribosomes attached to its surface synthesize proteins, which are then transported into the RER lumen for further processing and modification.

In contrast, the smooth endoplasmic reticulum (SER) lacks ribosomes on its surface. The SER is involved in lipid synthesis, detoxification, and other functions.