Complex sentence: The Barr-Root campaign promptly announced that it planned to appeal to the U.S. Supreme Court, but the court refused to hear the case, and on election day, Barr's name was absent from the state's ballot.
Split into simpler sentences: The Barr-Root campaign promptly announced that it planned to appeal to the U.S. Supreme Court. But the court refused to hear the case, and on election day, Barr's name was absent from the state's ballot.