Presidents can most directly influence the future decisions of the United States Supreme Court by

Responses

vetoing rulings to the Justices
vetoing rulings to the Justices

appointing new Justices to the Court with Senate approval
appointing new Justices to the Court with Senate approval

encouraging the public to write letters to the Justices
encouraging the public to write letters to the Justices

impeaching Justices with whom they disagree

impeaching Justices with whom they disagree is not a direct method of influencing future decisions of the Supreme Court. The correct answer is appointing new Justices to the Court with Senate approval. As Justices serve lifetime appointments, the opportunity to appoint new Justices when vacancies arise can have a significant impact on the ideological balance and direction of the Court for years to come.