Final answer:
The nurse's greatest concern when caring for a client receiving a thrombolytic agent after a myocardial infarction is pink frothy sputum, which can indicate pulmonary edema and compromised cardiac function.
Step-by-step explanation:
When caring for a client who is receiving a thrombolytic agent to open a clot occluded coronary artery after a myocardial infarction, the finding of greatest concern to the nurse would be Pink frothy sputum. Pink frothy sputum can indicate pulmonary edema, which is a serious complication of myocardial infarction and may signify a compromised cardiac function. Sero sanginous drainage from gums, hematemesis, and slight red color at urine are not directly related to the administration of a thrombolytic agent in the context of a myocardial infarction.