How can Voldemort trust Snape after the events of the first book?

Question

In the Sorcerer's Stone, Snape was playing with the Quirrell the double agent job. He was persuading him to find out how to get the stone.

"Have you found out how to get past that beast of Hagrid's yet?"
"B-b-but Severus, I --"
"You don't want me as your enemy, Quirrell," said Snape, taking a step toward him.
^{_{Harry-Potter-and-the-Philosopher's-Stone, CHAPTER THIRTEEN}}

But in the Quidditch match Snape

saved Harry and Quirrell knew that.

Later Harry discussed with Quirrell:

"But Snape tried to kill me!"
"No, no, no. I tried to kill you. Your friend Miss Granger accidentally knocked me over as she rushed to set fire to Snape at that Quidditch match. She broke my eye contact with you. Another few seconds and I'd have got you off that broom. I'd have managed it before then if Snape hadn't been muttering a countercurse, trying to save you."
"Snape was trying to save me?"
"Of course," said Quirrell coolly. (...)
^{_{Harry-Potter-and-the-Philosopher's-Stone, CHAPTER SEVENTEEN}}

How is it possible, that Voldemort trusted Snape later so much?

And that nobody brought this up. (I know, it may be out of screen, but Snape's position seems pretty strong)

Answer 1

All explained in the Half-Blood Prince (emphasis mine).

“Why did you stay there all that time, Snape? Still spying on Dumbledore for a master you believed dead?”

“Hardly,” said Snape, “although the Dark Lord is pleased that I never deserted my post: I had sixteen years of information on Dumbledore to give him when he returned, a rather more useful welcome-back present than endless reminiscences of how unpleasant Azkaban is…”

[...]

“But you didn’t return when he came back, you didn’t fly back to him at once when you felt the Dark Mark burn—”

“Correct. I returned two hours later. I returned on Dumbledore’s orders.”

“On Dumbledore’s—?” she began, in tones of outrage.

“Think!” said Snape, impatient again. “Think! By waiting two hours, just two hours, I ensured that I could remain at Hogwarts as a spy! By allowing Dumbledore to think that I was only returning to the Dark Lord’s side because I was ordered to, I have been able to pass information on Dumbledore and the Order of the Phoenix ever since! Consider, Bellatrix: The Dark Mark had been growing stronger for months. I knew he must be about to return, all the Death Eaters knew! I had plenty of time to think about what I wanted to do, to plan my next move, to escape like Karkaroff, didn’t I?

“The Dark Lord’s initial displeasure at my lateness vanished entirely, I assure you, when I explained that I remained faithful, although Dumbledore thought I was his man. Yes, the Dark Lord thought that I had left him forever, but he was wrong.”

_{^{Harry Potter and the Half-Blood Prince - Chapter Two, Spinner's End}}

So basically a combination of factors:

• Snape explained to Voldemort that he was ever faithful to him
• Snape had 16 years of information on Dumbledore to give to Voldemort
• Snape also had information on the Order to give to Voldemort
• Snape could still remain as Hogwarts as a spy (which also means even more information), owing to the fact that he didn't immediately apparate to Voldemort when the Dark Mark burned

---

Why does Voldemort still trust Snape if he failed to kill Harry?

“If I had murdered Harry Potter, the Dark Lord could not have used his blood to regenerate, making him invincible—”

[...]

“Indeed, many of the Dark Lords old followers thought Potter might be a standard around which we could all rally once more. I was curious, I admit it, and not at all inclined to murder him the moment he set foot in the castle.

Answer 2

The reason for this is explicitly mentioned in HBP as to why he stood between Quirrell and the stone.

“I think you next wanted to know,” he pressed on, a little more loudly, for Bellatrix showed every sign of interrupting, “why I stood between the Dark Lord and the Philosopher's Stone. That is easily answered. He did not know whether he could trust me. He thought, like you, that I had turned from faithful Death Eater to Dumbledore’s stooge. He was in a pitiable condition, very weak, sharing the body of a mediocre wizard. He did not dare reveal himself to a former ally if that ally might turn him over to Dumbledore or the Ministry. I deeply regret that he did not trust me. He would have returned to power three years sooner. As it was, I saw only greedy and unworthy Quirrell attempting to steal the stone and, I admit, I did all I could to thwart him.”
_{(Half-Blood Prince, Chapter 2, Spinner's End).}

Answer 3

"I saw you and Snape in the Forest –" he blurted out.
"Yes," said Quirrell idly, walking around the Mirror to look at the back. "He was on to me by that time, trying to find out how far I'd got. He suspected me all along. Tried to frighten me – as though he could, when I had Lord Voldemort on my side ..."

It's not quite sure if Snape suspected Quirrell to be possessed. Voldemort may not have expected Snape to know, either.

So while Snape saved Harry and Voldemort knew he did, Voldemort may have well assumed that he did so not knowing Quirrell was trying to kill Harry on Voldemort's direct orders.
Given that Harry's prolonged existence allowed Voldemort to return to the flesh more powerful then before, he might not have minded Snape keeping the boy alive until then.