Quadratic pair search

This function finds whether any two distinct elements of nums add up to target. It times out for n = 10⁵. Make it fast enough, and discuss the trade-offs of the two standard approaches.

1def has_pair(nums, target):
2    n = len(nums)
3    for i in range(n):
4        for j in range(i + 1, n):
5            if nums[i] + nums[j] == target:
6                return True
7    return False

1. Explain why n = 10⁵ times out: how many pair checks are made?
2. Identify the redundant work: for a fixed nums[i], what single question are we asking about the rest of the array?
3. Give an O(n) solution and an O(n log n) O(1)-extra-space solution.
4. Discuss: which one returns indices easily, which one handles duplicates and the x + x == target case, which works on a stream?