。
She turned up her nose at our offer.
Literal
She [topic-は] our offer [object-を] [with-で nose] brushed-off.
鼻であしらう is a vivid set idiom — literally 'to handle [something] with the nose,' meaning to dismiss contemptuously, with disdain. The image is of someone wrinkling their nose at a proposal too lowly to deserve a real response. あしらう itself ('to handle, to deal with') is a relatively formal verb; combined with 鼻で it becomes a fixed expression. 申し出 ('proposal, offer, suggestion') is the noun form of 申し出る ('to offer, to propose') — a polite Sino-Japanese reading.