Passage
Their sorrows shall be multiplied that give gifts for another [god]: Their drink-offerings of blood will I not offer, Nor take their names upon my lips.
Psalms 16:4 (ASV)
Nearby Context
Psalms 16:2 [O my soul], thou hast said unto Jehovah, Thou art my Lord: I have no good beyond thee.
Psalms 16:3 As for the saints that are in the earth, They are the excellent in whom is all my delight.
Psalms 16:4 Their sorrows shall be multiplied that give gifts for another [god]: Their drink-offerings of blood will I not offer, Nor take their names upon my lips.
Psalms 16:5 Jehovah is the portion of mine inheritance and of my cup: Thou maintainest my lot.
Psalms 16:6 The lines are fallen unto me in pleasant places; Yea, I have a goodly heritage.
Study Lenses
The verse centers on "sorrows", "shall", "multiplied", "give", "gifts", "another", "drink-offerings", and "blood". It is saying that the verse draws attention to "sorrows" and "shall", so its meaning should be read from those terms before moving to application.
The nearby context moves from verse 3's "As for the saints that are in..." into verse 5's "Jehovah is the portion of mine inheritance...", so "sorrows" and "shall" belong inside that flow. In Psalms context, the local focus is worship, trust, the LORD's kingship, and covenant mercy.
A plain takeaway is to answer the verse's own emphasis on "sorrows" and "shall" with trust shaped by these words, not by a vague optimism outside the passage.