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the only verb there is "bid"
adieu is a noun or interjection
past tense of bid is "bade"
ergo
I bade him adieu.
No, "bided adieu" is not correct. The correct phrase is "bade adieu," which means to say goodbye or farewell.
I bid you adieu.
Perfectly fine "I'll be happy to bid adieu to these braces" "Well, friends, it's time I bid you all adieu; I've got a 6:10 tee-time tomorrow"
good bye