Choose the sentence where the idiom has been incorrectly used.

The correct answer is Option (3) → She kept out talking.

A phrasal verb is a verb joined with a small word such as up, out or together. The two words together have a new meaning.

Keep out means to stay outside, or not to enter ("Keep out of the kitchen"). To say that someone went on talking, the phrasal verb is keep on + -ing, meaning to continue doing something. The correct sentence is "She kept on talking."
Example: The baby kept on crying all night.

Why the Other Options are Incorrect?

Option 1: Please hear me out. — Incorrect. Correct use: hear someone out means to listen to someone until he has finished speaking.
Example: Don't interrupt. Hear him out first.

Option 2: The officers agreed to carry out the orders. — Incorrect. Correct use: carry out means to do or complete a task, an order or a plan.
Example: The scientists carried out a series of tests.

Option 4: I look after my parents. — Incorrect. Correct use: look after means to take care of someone.
Example: My aunt looks after the children when we travel.