The present participle of bedrivel, meaning to cover or make messy with drivel (saliva or nonsense).
From the prefix 'be-' (meaning 'to cause to be') combined with 'drivel' (from Middle English drivelen, possibly of Scandinavian origin meaning to flow or drip). The form became increasingly archaic through the 17th-19th centuries.
The 'be-' prefix was incredibly productive in English, creating verbs that mean 'to cause something to become X' — bedrivel, befuddle, besmirch — but most of these colorful words have faded from modern use, leaving us with a less vivid vocabulary.
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