。
She answered every question with confidence.
Literal
She [topic-は] all [possessive-の] questions [to-に] confidence [object-を] holding answered.
自信を持って literally means 'holding confidence' — the te-form 持って functions adverbially, describing the manner of answering. This is a productive pattern: a te-form phrase preceding a main verb modifies how the action is performed. 全て (すべて) is a slightly more formal or elegant way to say 'all' compared to 全部.