In Hamlet, what explanation does Gertrude offer for Polonius' murder?
What Gertrude leaves out is more important than what she tells Claudius. She does not tell him that Hamlet stabbed Polonius through the arras because the old man was yelling for the guards. Why was he yelling for the guards? Because Gertrude thought Hamlet was going to murder her and she was calling for the guards. Hamlet would not even have known that Polonius was there in hiding if the old man hadn't started shouting,
What, ho! Help, help, help! III.4
Gertrude tells Claudius:
In his lawless fit,
Behind the arras hearing something stir,
Whips out his rapier, cries, "A rat, a rat!"
And in this brainish apprehension kills
The unseen good old man. IV.1
She deliberately lies to her husband to protect her son. She emphasizes that he is mad, although she now knows he is not. She gives Claudius an entirely false description of Hamlet's reaction upon finding he has killed Polonius. Claudius asks her, "Where is he gone?" and she replies:
To draw apart the body he hath killed,
O'er whom his very madness, like some ore
Among a mineral of metals base,
Shows itself pure; he weeps for what is done. IV.1
Hamlet doesn't show any remorse at all. He sees he has killed Polonius and says:
Thou wretched, rash, intruding fool, farewell!
I took thee for thy better. Take thy fortune.
Thou find'st to be too busy in some danger. III.4
Gertrude intentionally hides the whole substance of her meeting with her son. She intended to upbraid him for offending her husband at the play within a play. Instead he vents all his pent-up anger at her for forgetting her first husband so easily and committing what he considers incest and adultery. He reduces her to tears and reverses their relationship. Previously she had been the authoritative parent, but he has now become the dominant one. This violent scene marks a turning point in the play. Hamlet will no longer be passive and vacillating but forceful and determined. However, he will still continue to play mind-games with Claudius, pretending to be totally mad now that he has killed Polonius.
Gertrude thought Hamlet intended to kill her, and she knew he thought he was killing Claudius himself when he stabbed Polonius through the arras. This is what she withholds from her husband. If Claudius knew about Hamlet's true intentions he would have to take strong action against him. As he says to Gertrude when she tells him how Hamlet killed Polonius:
O heavy deed!
It had been so with us, had we been there. IV.1
It should be noted that what starts the whole imbroglio is a metaphor. Hamlet tells his mother:
Come, come, and sit you down. You shall not budge.
You go not till I set you up a glass
Where you may see the inmost part of you. III.4
And she panics:
What wilt thou do? thou will not murder me? Help, ho! III.4
Gertrude believes her son is really insane. She thinks Hamlet means to set up a mirror and cut her open with his rapier and force her to look at her own bleeding intestines—whereas he was only speaking metaphorically. She screams for help. Polonius can't see what is happening and starts screaming too. Hamlet thinks he has walked into a trap, a "set-up," and that both his mother and the man behind the arras (possibly Claudius) may intend to have him thrown into a dungeon permanently. So he kills the old man in self-defense. This bloody deed so unhinges Gertrude that she is completely submissive to her son throughout the rest of the scene. He has a bloody rapier in his hand, and she can't be sure he won't use it on her too! It is excellent stagecraft.
The queen gives a very simple reason: her son is insane. She describes him as being "Mad as the sea and wind, when both contend which is mightier" (IV.i.7-8). She is telling this to the king though, so who knows if she really believes Hamlet is mad, or just telling the king he is mad to protect him.
She then goes on to describe how Hamlet wept over his murder, saying that it was as if "ore among a mineral of metals base, shows itself pure"(IV.i.25-27). The queen compares Hamlet's madness to a base (dirty) mineral, and his show of compassion towards Polonius' death an ore, pure and beautiful. This contradictory behavior from Hamlet just confuses her more, and almost convinces her that her son really has lost his grip on reality.