Question

Which sentence uses correct punctuation?

After our talk, Karen spoke of her hope: “I intend to run for student council and win.’
After our talk, “Karen spoke of her hope: I intend to run for student council and win.”
After our talk, Karen spoke of her hope: “I intend to run for student council and win.”
After our talk, Karen spoke of her hope: I intend to run for student council and win.

Answer 1

The third one is correct because when somebody speaks in a sentence, you have to put in in quotation marks on both sides, the other sentences do not have proper quotation marks.... ( " ") 

Answer 2

The answer to your question would be that the sentence that uses correct punctuation is the following one: After our talk, Karen spoke of her hope: "I intend to run for student council and win."

What is enclosed in the quotes should be the exact words said by Karen in this case. Anything that is not part of those exact quotes must be placed outside the quotes. What is more, colons and semicolons go outside of the final quotation mark. That is why the third option is the correct one.