Aleksandrov, Kolmogorov and Lavrent'ev state that x⁵ + x - a is nonsolvable for a = 3,4,5,7,8,9,10,11,... . In other words, these polynomials have a nonsolvable Galois group. A full explanation of this sequence requires consideration of both reducible and irreducible solvable quintic polynomials of the form x⁵ + x - a. All omissions from this sequence due to solvability are characterized. This requires the determination of the rational points on a genus 3 curve.