If by direct or by collateral hand they find us touch'd, we will our kingdom give, our crown, our life, and all that we call ours, to you in satisfaction; but if not, be you content to lend your patience to us, and we shall jointly labour with your soul to give it due content. how does king claudius's speech relate to the central conflict in hamlet?

The promise made by King Claudius to Laertes to help avenge the death of his father Polonius serves as another factor for the plot of the story to move forward.

Explanation:

The central conflict in William Shakespeare's tragic play "Hamlet" is the attempts of the young prince Hamlet to avenge the death of his father King Hamlet. Likewise, the unintended death of Polonius at the hands of Prince Hamlet also led to the present King and main suspect in King Hamlet's murder, Claudius to have a reason to push forward the lot of the play.

With the promise of revenge in his mind, he provide another aspect for the plot to advance further. Knowing he needed to keep his promise to Polonius' son Laertes, Claudius also begins another story line for his primary idea of getting rid of Hamlet. Thus, the speech of Claudius in Act IV scene v acts as another source for the further advancement of the plot.