In a lengthy sentence consisting of many phrases, it is essential to determine which phrases and words are necessary to the sentence and which words may be eliminated because they are unnecessary. The relative pronoun that correctly refers to programs and introduces the subordinate clause;family is followed by two phrases that are clear and correct. To be able to move, however, is needlessly wordy, repeating the meaning of enable, and can be reduced to 'to move'. This creates a parallel construction in which programs ... enable a family ... to move ... and to apply
A To be able to move is wordy and able is redundant after enable; to apply is not logically parallel to the infinitive phrase (able) to move.
B Correct. In this sentence, eliminating the wordy construction to be able allows to move to be parallel to to apply.
C Insofar as this is grammatical, using a semicolon here causes that to refer too broadly to the entire previous clause rather than specifically to programs; the two infinitives should be joined by the conjunction and, not separated by a comma.
D Enables does not agree with the plural subject; applying following a nonrestrictive clause suggests incorrectly that the builders not the family, are applying the rent.
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E The comma after programs is incorrect because the clause is meant to be restrictive' , as in D, applying will alter the meaning of the sentence.