Why does Macbeth order the murder of Banquo and his son Fleance in Macbeth?
In act 3, scene 1 of William Shakespeare's Macbeth, Macbeth explains the reason why he intends to order the murder of Banquo, his onetime friend and comrade-in-arms, and Banquo's son, Fleance.
MACBETH. To be thus is nothing,
But to be safely thus. (3.1.52–53)
Macbeth's priority up to this point in the play has been "To be thus" —that is, to become king. Now that Macbeth is King, his new priority is "to be safely thus"—to remain king.
Macbeth hasn't forgotten the prophecy that the witches made to Banquo at the same time that they prophesized that Macbeth "shalt be King hereafter!" (1.3.52).
THIRD WITCH. [to Banquo] Thou shalt get kings, though thou be none. (1.3.70)
This prophecy troubles Macbeth to such an extent that he mentions it twice to Banquo in that scene: once almost immediately after the prophecy is made, "Your children shall be kings" (1.3.89), and once again after Ross and Angus tell Macbeth that King Duncan made him Thane of Cawdor, fulfilling one of the prophecies that the witches made to Macbeth.
MACBETH. Do you not hope your children shall be kings,
When those that gave the Thane of Cawdor to me
Promised no less to them? (1.3.127–129)
Macbeth seems more concerned about Banquo's children being kings than with being king himself.
The prophecy isn't mentioned again until the beginning of act 3, scene 1, when Banquo himself raises the issue. Banquo thinks aloud about his suspicions that Macbeth murdered Duncan, "and I fear / Thou play'd most foully for't" (3.1.2–3), and he also notes that the witches said that no descendant of Macbeth will be king, "But that myself should be the root and father / Of many kings" (3.1.5–6).
With Banquo's short speech at the beginning of the act, Shakespeare tells the audience that Macbeth has two reasons to fear Banquo. The first reason is that Banquo suspects that Macbeth murdered Duncan, and the second is that Banquo's descendants, not his own, will succeed him as king. It turns out that Macbeth is much more concerned with Banquo's descendants than with Banquo's suspicions.
Macbeth enters the scene and invites Banquo to his coronation feast, then casually inquires, "Ride you this afternoon?" (3.1.21). A few lines later, Macbeth asks, "Is't far you ride?" (3.1.26).
Macbeth knows that Banquo suspects his involvement in Duncan's death, and he weakly tries to deflect that suspicion by mentioning to Banquo that Duncan's sons, Malcolm and Donalbain, have fled Scotland, "not confessing their cruel parricide" (3.1.34–35). Then Macbeth asks, "Goes Fleance with you?" (3.1.39).
The reason for these incidental questions soon becomes clear. Macbeth engages two murderers to kill Banquo, "and with him... Fleance his son," while they're riding outside the castle.
Until Macbeth can rid himself of Banquo and Fleance, Macbeth wears a "fruitless crown" and carries a "barren sceptre" (3.1.65–66) and will continue to fear the prophecy that Banquo's descendants will be kings.