Wouldn't the the form of the particular solution be just ?
The book says, if right side is , then you should try
where is the number of solutions to the charateristic solution that is equal to .
And I think, we always have on the right side to to speak.
They do something like that in a previous example.
Edit: I tried your suggestion, it worked. I guess I misinterpreted the book. Here we only have , so we donīt mulitply by any .