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The South resisted the decision of Brown vs. Board of Education. This makes sense. Brown vs. Board of Education ruled that segregating schools based on race was unconstitutional.
This line is foreshadowing because it implies that anything -- even people -- that becomes a burden can easily be destroyed.
Later, when Beatty himself becomes a problem, Montag burns Beatty to death. When Montag killed Beatty, he destroyed the responsibility he owed him. In this way, this line is an element of foreshadowing.