Intro
Just like King Lear, Shakespeare's Hamlet exists in several versions: the First Quarto version, the Second Quarto version, and the Folio version. The First Quarto version is widely believed to be the play as reconstructed from memory by an actor in Shakespeare's company who performed in the play. The Folio version, even though it was compiled by Shakespeare's buddies after his death, is believed to be the most authoritative.
Let's zoom in on one of Hamlet's famous soliloquies in Act 1. He's raving about a lot of things here, and especially about his mom's marriage to Claudius, a sneaky dude who killed Hamlet's father and took over the throne. We'll look at two versions of the soliloquy: from the First Quarto and the Folio.
Quote
Quarto 1
Ham. O that this too much grieu'd and sallied flesh
Would melt to nothing, or that the vniuersall
Globe of heauen would turne al to a Chaos!
O God, within two months; no not two: married,
Mine vncle: O let me not thinke of it,
My fathers brother: but no more like
My father, then I to Hercules.
Within two months, ere yet the salt of most
Vnrighteous teares had left their flushing
In her galled eyes: she married, O God, a beast
Deuoyd of reason would not haue made
Such speede: Frailtie, thy name is Woman,
Why she would hang on him, as if increase
Of appetite had growne by what it looked on.
O wicked wicked speede, to make such
Dexteritie to incestuous sheetes,
Folio
Ham. Oh that this too too solid Flesh, would melt,
Thaw, and resolue it selfe into a Dew:
Or that the Euerlasting had not fixt
His Cannon 'gainst Selfe-slaughter. O God, O God!
How weary, stale, flat, and vnprofitable
Seemes to me all the vses of this world?
Fie on't? Oh fie, fie, 'tis an vnweeded Garden
That growes to Seed: Things rank, and grosse in Nature
Possesse it meerely. That it should come to this:
But two months dead: Nay, not so much; not two,
So excellent a King, that was to this
Hiperion to a Satyre: so louing to my Mother,
That he might not beteene the windes of heauen
Visit her face too roughly. Heauen and Earth
Must I remember: why she would hang on him,
As if encrease of Appetite had growne
By what it fed on; and yet within a month?
Let me not thinke on't: Frailty, thy name is woman.
Analysis
Notice any differences? Let's start with the obvious things—like the fact that the soliloquy in the First Quarto is much, much shorter than the soliloquy in the Folio.
And let's be real: the Folio soliloquy is just way more eloquent and beautiful than the soliloquy in the Quarto. In the Quarto, for example, Hamlet says: "O that this too much grieu'd and sallied flesh / Would melt to nothing, or that the vniuersall / Globe of heauen would turne al to a Chaos!"
In the Folio, this becomes: "Oh that this too too solid Flesh, would melt, / Thaw, and resolue it selfe into a Dew: / Or that the Euerlasting had not fixt / His Cannon 'gainst Selfe-slaughter." The Folio version is more dramatic, it flows betters, and it has more vivid and striking imagery. It's just stronger in general.
The point is that just by comparing and contrasting these two versions of the soliloquy, Textual Critics can arrive at some very important conclusions. One of the reasons scholars think the Quarto version of the play is reconstructed from an actor's memory is that there are big gaps, even within a single soliloquy.
We can also assume that the Folio version is closer to Shakespeare's language because, hey, it's way more beautiful and detailed. We're talking about Shakespeare here: the dude totes knew how to put words together. And since the words are put together better in the Folio soliloquy than the Quarto soliloquy, we can pretty safely conclude that the Folio soliloquy is the more "authoritative" of the two versions.