。
She develops her arguments based on facts.
Literal
She [topic-は] facts [on-に] basing, argument [object-を] develops.
~に基づいて ('based on, founded on') is a formal compound conjunction — the te-form of 基づく ('to be based on'), with 基 ('foundation') as the kanji root. Common in academic, legal, and journalistic prose. 議論を展開する ('develop / unfold an argument') uses 展開 ('development, unfolding') as a noun verbalized with する — a Sino-Japanese compound with strong intellectual flavor, used for arguments, plots, business strategies, and storylines. Together: a sentence that would fit comfortably in a news editorial or academic abstract.