Which condition may lead to the expulsion of blood through the cough?

The term that refers to the expulsion of blood through coughing is "hemoptysis." This condition specifically involves the coughing up of blood or blood-stained sputum, which usually originates from the respiratory tract, typically the lungs or bronchial tubes. The presence of blood in the respiratory secretions can indicate various underlying health issues, including infections, tumors, or vascular abnormalities.

The other choices refer to distinct medical conditions. "Hematocele" involves a collection of blood in a cavity, commonly in relation to the testicles; "hemiplegia" refers to paralysis of one side of the body, usually due to a stroke; and "hematochezia" pertains to the passage of fresh blood through the anus, indicating bleeding from the lower gastrointestinal tract. None of these conditions involve coughing up blood, which is why they are not relevant to the question.