[take] a complaint

to accept or receive an expression of dissatisfaction or criticism from a customer, client, etc. regarding a product, service, or experience

As a customer service representative, she is trained to take complaints and address them efficiently.
He took the customer's complaint seriously and promised to investigate the issue.
Taking complaints into consideration can help improve the quality of a business.
The restaurant manager took the complaint about the food quality and offered a complimentary meal.
The hotel's front desk staff is available to take a complaint from any guest with concerns about their stay.