explain how the manner in which the Vietnam war was fought led to defeat for the US and South vietnam

Viet Nam was fought with the doctrine that had evolved from the Korean War of fighting a limited land war in Asia. This shaped the most basic military principles, which prevented military action that would have crippled the military ability of the North. This ended up being a recipe for failure, as the NVA were resilient despite the punishment meted out by vastly superior US firepower.

The other key political error was failing to recognize how the political situation in the South undermined any military effort - that the government that the US was fighting to save was a corrupt leftover of French colonialism, and had no support. It was replaced by a military coup that had even less support. The communist North represented the Vietnamese people and their nationalist desire for independence from outside rule. It is unfortunate that in that era in many places, the communists represented what most of the locals wanted, which was freedom from colonial oppression, and were also the most effective force fighting for that freedom. The US policy of anti-communism ended up putting the US on the side of corruption and the imperialism that had built colonial empires, and was fighting against the desire for freedom by the locals. It was not a winning political strategy.

The situation in South Viet Nam was more complicated than that, but compare this analysis with the domino theory that was prevalent in shaping US decision making. It meant that any expediency in order to resist communism made sense, which meant ignoring the hearts and minds of the locals in making decisions. This political thinking was criticized at the time in the US, but even in later decades, the US justified support of right wing dictatorships just as evil as any communist dictatorship based on the fact that it was anti-communist. This failure of political thinking undermined any meaningful counter insurgency doctrine.