Complex sentence: Dewey played a key role in convincing Eisenhower to run against Taft, and when Eisenhower became a candidate Dewey used his powerful political machine to win Eisenhower the support of delegates in New York and elsewhere.
Split into simpler sentences: Dewey played a key role in convincing Eisenhower to run against Taft. When Eisenhower became a candidate Dewey used his powerful political machine to win Eisenhower the support of delegates in New York and elsewhere.