Though Grendel is dead, Grendel's mother still lives, and wants revenge for the death of her son. She enters Heorot and seizes a man, waking the other warriors. Frightened, she takes Grendel's arm from its place under the roof, and flees.
Beowulf, who had slept in a private chamber that night, is brought to Heorot. Hrothgar tells him that Grendel grabbed Aeschere, Hrothgar's adviser and companion in battle. Hrothgar adds that he has heard of two huge creatures walking on the moors and wastelands. One of these creatures looked like a woman, the other looked like a man. These two monsters dwell in the dark, in a lake so deep it seems bottomless. Hrothgar says he must depend on Beowulf a second time, and offers him treasure to kill Grendel's mother and end the feud.
As a loyal lord, it's Hrothgar's obligation to avenge Aeschere's death. So now Hrothgar seeks revenge, just as Grendel's mother does. This is how feuds are born, and continued, until one people is thoroughly beaten by another.
Beowulf tells Hrothgar: "It is better for a man to avenge his friend than to mourn too much." He adds that death comes to everyone, and then suggests that they follow Grendel's mother back to her lair immediately.
Note the contrast between the king, Hrothgar, thinking of his people, and Beowulf, the warrior, thinking of battle and great deeds.