Explanation

n² + 19n + 92 = k² gives (2n + 19)² − (2k)² = −7, i.e. (2k − 2n − 19)(2k + 2n + 19) = 7. The integer factor pairs of 7 are (1,7), (7,1), (−1,−7), (−7,−1); both factors are odd so all four pairs are admissible. They give 2n + 19 = ±3, i.e. n = −8 or n = −11. The sum is −19. The answer is (D).