Final answer:
The democratic leadership style is not included in House's path-goal theory; the theory's styles are supportive, directive, participative, and achievement-oriented leadership.
Step-by-step explanation:
The leadership style that is NOT one of the four leadership styles in Robert House's path-goal theory is democratic. House's path-goal theory identifies four styles of leadership which are supportive leadership, directive leadership, participative leadership, and achievement-oriented leadership. The democratic leadership style, although widely recognized in various contexts of group and organizational behavior, is not part of this particular theory. Instead, the democratic leadership style is characterized by a leader who encourages group participation in decision-making, consensus-building, and is often used in settings like clubs or teams where members have a say in the activities and projects pursued.