"Oh good, now you can give that graffitied copy back."
-- Hermione Granger (HBP11)
Advanced Potion-Making is a potions textbook used at Hogwarts in the 1990s.
The N.E.W.T.-level textbook for Potions during Harry's sixth year (HBP9), which cost nine Galleons brand-new (HBP11). The book was written around the year 1946 (HBP16). The book contains instructions for making the Draught of Living Death starting at page 10 (HBP9).
The Half-Blood Prince's copy of this book, which Harry borrowed from Professor Slughorn, was annotated with many useful additions throughout, enabling Harry to earn a reputation as a very skilled potion-maker. Among the Half-Blood Prince's additional notes were the invented spells Levicorpus and Sectumsempra, and the instructions to 'just shove a bezoar down their throats' in case of a poisoning.
The Half-Blood Prince's copy of this book was probably destroyed when Fiendfyre tore through the Room of Requirement during the Battle of Hogwarts.
The author's first name is likely a play on the word 'libation' meaning a drink, particularly one drunk or given as part of a ceremony or ritual. Borage is a type of herb.
From the Web
"Advanced Potion Making" on the Harry Potter Wiki