Sky Scythe Broom:
Starting at the East Hogsmeade Valley floo flame, head outside and go directly toward the tent in front of you, and talk with Leopold Babcocke.
The Sky Scythe Broom will be the first item available in the shop.
Silver Arrow Broom:
Next, fast travel to the Lower Hogsfield floo flame and talk with Arn. He’ll tell you that Ranrok’s Loyalist stole his carts, and have them at an encampment nearby, starting the Carted Away side quest.
Now you just need to follow the path south, using the waypoint on the mini-map until you come across the goblin encampment.
You’ll need to clear out all the goblins in the area before being allowed to open the gate freeing the carts, which will now make their way back to Arn.
Now head back and talk to Arn, finishing the quest, which unlocks his shop just south of Lower Hogsfield.
Once here, just talk with him, and the Silver Arrow Broom will be the first item available in the shop.
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