Final answer:
Frost uses the extended metaphor of comparing a patch of old snow to sleep in his poem.
Step-by-step explanation:
The extended metaphor that Frost uses throughout his poem is option B, where he compares a patch of old snow to sleep. This metaphor is present in the lines "The woods are lovely, dark and deep, / But I have promises to keep, / And miles to go before I sleep." Here, Frost compares the quiet and peacefulness of the snowy woods to the serenity of sleep, suggesting that the speaker is reluctant to give in to the temptation of rest and instead must continue on his journey.