Harry, Ron, and Hermione receive a tear-stained note from Hagrid telling them that Aragog, his giant spider friend, has died. Hagrid is clearly upset and asks them to come help him bury his good friend.
The three aren't too keen on the idea, considering it would be violating Hogwarts rules to sneak out of the school walls after dark, and they're busy with plenty of other things. They have a hard time understanding how Hagrid could've loved a freaky, giant spider so much.
Ron and Hermione take their Apparition test. Harry is not yet seventeen, and so he isn't qualified to take the test yet. Because most of his classmates are taking the test, Harry is one of the few students in Potions class that day.
As soon as Potions ends, Harry tries again to ask Slughorn for the Horcrux memory, but to no avail.
Harry returns to the common room to discover that Hermione has passed her Apparition test, but Ron has not due to an eyebrow that got left behind while he was Apparating.
Hermione helps Harry strategize about how best to get the Horcrux memory from Slughorn, and Ron has the brilliant idea that Harry should drink a bit of his Felix Felicis (good luck potion) and to seek out the memory that way.
Once Harry drinks a bit of the potion, he knows what he must do: he has to go to Hagrid's house and help him bury Aragog. This seems like a bad plan to Hermione and Ron, but Harry knows in his gut that this is the right path to take.
On his way to Hagrid's, Harry runs into Slughorn who agrees to come with Harry and help bury Aragog (Aragog's venom is worth lots of Galleons, and Slughorn can't resist).
Hagrid is delighted to see Harry and Slughorn.
After a nice burial, the three retire to Hagrid's house where they sit around the table and tell stories to pass the time and to cheer up Hagrid.
Lots of wine is shared, but Harry knows not to drink a drop but to remain totally coherent and sober.
As Slughorn and Hagrid become more and more sloshed and jovial, Harry begins to ask Slughorn for the Horcrux memory.
Slughorn is at first reluctant, but when Harry tells him that he needs the memory because he has to fight Voldemort, Slughorn gives in.
We learn that the crux of Slughorn's reluctance to give up the Horcrux memory lies in the fact that he has huge guilt about what it reveals about him.