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She washed her dirty hands before eating.
Literal
She [topic-は] meal [possessive-の] before [to-に] dirty hands [object-を] washed.
食事の前に ('before the meal') uses the noun-based ~の前に pattern. 汚れた (よごれた, 'dirty, soiled') is the past tense of 汚れる ('to become dirty') used as an adjective — a common pattern where the past form of a change-of-state verb describes the resulting state.