The teacher asked those who completed their test __ the room as quickly as possible .

选 A
这里考察点是ask sb to do sth，让某人做某事。
这句话的意思是老师让那些完成试题的人尽快离开课室。
其实这里只是在those后面加了定语从句who completed their test ,只要把这个从句去掉你就看得很清晰了，那样就变成The teacher asked those to leave the room as quickly as possible

to leave
实际上是两个并列的句子。ask to do st.

A to leave
ask sb. to do sth

A ask sb to do sth
老师叫那些完成考试的学生尽快离开考场.

A
ask sb to do sth.要求某人做某事
老师让那些答完卷的同学尽快离开房间.

to leave 整个句子是个定语从句
其实是The teacher asked those to leave the room as quickly as possible 和 the people completed their test 还 搭配后的定语从句
Ask sb. to do sth. 这个句型