Complex sentence: Ralph Nader's name appeared in the press as a potential candidate for president for the first time in 1971, when he was offered the opportunity to run as the presidential candidate for the New Party, a progressive split-off from the Democratic Party in 1972.
Split into simpler sentences: Ralph Nader's name appeared in the press as a potential candidate for president starting in 1971. Nader was offered the opportunity to run as the presidential candidate for the New Party, a progressive split-off from the Democratic Party in 1972.