Final answer:
The colon should be placed after 'something' to introduce the list of achievements: "Luis felt that he had accomplished something: he had written and recorded his first song."
Step-by-step explanation:
The question about where to place a colon in the sentence, "Luis felt that he had accomplished something he had written and recorded his first song," implies adding punctuations for clarity. An appropriate way to use a colon here would be to introduce a list, a quote, an explanation, or a consequence.
Since Luis's accomplishments are listed after the initial clause, the colon should go after the word 'something': "Luis felt that he had accomplished something: he had written and recorded his first song." The colon emphasizes that what follows is a detailed explanation of what Luis felt he had accomplished.