A 28-year-old man shot and killed his girlfriend inside their apartment in Federal Way, south of Seattle, then went on a rampage that ended in his own death and that of three other men, the authorities said. Police Chief Brian Wilson of Federal Way said Monday that officers investigating a 911 call at the apartment were confronted in the stairway by the man with a shotgun. They fired at him, Chief Wilson said, but the man fled to a nearby parking lot. He said they found the man on the ground, and opened fire when he reached for a handgun. He died at the scene. Neither the suspect nor the victims have been identified. The police believe the suspect shot his girlfriend and left their apartment. He then argued with two men and fatally shot them both, authorities said. A third man went outside to investigate then retreated inside his residence after urging others to call 911. Chief Wilson said the suspect broke down that man’s door with a shotgun and killed him.