Final answer:
Eukaryotic cells expel materials like hormones to the extracellular space through exocytosis, where vesicles fuse with the plasma membrane to release their contents outside the cell.
Step-by-step explanation:
Eukaryotic cells release material out of the cell through a process called exocytosis. An example of materials commonly released this way is hormones. Vesicles containing substances, such as digestive enzymes, histamine, or hormones, fuse with the plasma membrane and the contents are ejected into the extracellular space. This process is essential for various physiological functions, including the secretion of neurotransmitters at synapses between neurons and the production of components of the extracellular matrix.