How did the united states administer its newfound
territories?

This is in relation to the territories acquired by the United States from the Spanish after the Spanish-American War. The territories collected and their administration were:

After Guam was taken over by the United States, the President issued  Executive Order 108-A which transferred administration of the islands to the Navy.

With Puerto Rico, the President allowed the U.S. military to take control and would then appoint a governor-general. Puerto Rico was allowed a House of Representatives but the U.S. would appoint the Upper House and the governor.

The Philippines were not very easy to control and the U.S. had to defeat them in war and consequently maintain a strong presence. The Philippines had a civilian government as the governor-general and the Upper House of Legislature were appointed by the U.S..

In conclusion, the newly acquired territories were in some cases given some independence but were still firmly under the control of the U.S.

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The correct answer to this open question is the following.

The United States administered its newfound territories in the following way.

We are talking about the new territories that the United States gained after defeating Spain in the Spanish-American War of 1898. The new territories were Guam, the Philippines, and Puerto Rico. Guam became an incorporated territory for the United States in Micronesia. Puerto Rico became a protectorate of the United States. The Philippines had many conflicts and tried to rebel against the control of the US, and finally became independent. In the case of Cuba, it never was part of the US. It became independent from Spain but always had a heavy influenced by the US until Cuba's Revolution that was led by Fidel Castro.

the main reason that the constitution did not proclaim all men to be free and equal is because of the issue of slavery.