Final answer:
A reliable method for assessing correct ET tube placement in a child is using a combination of direct visualization, auscultation with a stethoscope, and monitoring with a pulse oximeter to ensure accurate positioning and adequate oxygenation.
Step-by-step explanation:
The method that provides a reliable, prompt assessment of correct endotracheal (ET) tube placement in a child who has undergone elective ET intubation due to respiratory failure includes a combination of direct visualization, auscultation with a stethoscope, and the use of a pulse oximeter. Direct visualization during intubation ensures the tube passes through the vocal cords. Immediately after placement, auscultation over the chest is used to listen for bilateral breath sounds, while absence of sounds over the stomach indicates the tube is not in the esophagus. The pulse oximeter is helpful to monitor oxygen saturation and can indicate adequate ventilation if the readings improve or stabilize.