After listening to the king who explained to Beowulf the location and existence of Grendel’s mother and lamenting about the death of his friend. Beowulf asked the king to stop mourning his friend and instead avenge him.He also added that each man should strive for glory before they pass on.
“Do not lament, wise sire! It seems better that each man avenge his friends than to mourn them to no end. Each of us must await the end of his path in this world, and he who can, should achieve renown before death! That is the best memorial when life is past and a warrior's days are recounted. Rise up, oh warden of the realm!
We ride forth promptly to catch the trail of Grendel's mother.
Mark my words—she shall find no shelter, neither in the earth of the fields, nor the mountainous woods, nor the ocean's depths—wherever she may flee! Have patience and endure your woes this day, as I suspect you shall.”
According to Beowulf, it was the glorious deeds achieved when one is alive that will be remembered when they have passed on. He asked Hrothgar to bear the pain of his loss and promised to seek vengeance for him by pursuing Grendel’s mother where she lived.
Beowulf later pursues the monster and challenges her to a vicious battle in her lair. The warrior succeeds in killing Grendel’s mother. He also takes Grendel's head, and the hilt of the sword used to kill Grendel’s mother and presents them to Hrothgar.