What phrase came to define the idea that whites should civilize
nonwhites?

The phrase that came to define the idea that whites should help civilize nonwhites was "White Man's Burden".

Explanation:

"The White Man's Burden" was an ideological concept in the service of Western imperialism. It was one of the dominant ideological and moral motivations of Western empires as they conquered much of Africa and Asia and other parts of the world in the 19th century. The phrase appears for the first time in the poem "The White Man's Burden" by English poet Rudyard Kipling for Queen Victoria's jubilee in order to celebrate and encourage the extension of the British Empire. The poem was published in McClure's magazine in 1899 in the United States, with the subtitle 'The United States and the Philippine Islands'. The phrase was popularized by imperialists in the USA as a justification for imperialism in moral terms.

According to the viewpoint summarized in the phrase, it was the "White Man's Burden" to civilize the "savages" outside the West. Behind the idea was the belief that whites were more advanced and civilized than indigenous peoples and their moral duty was to "elevate" them and "enlighten" them according to their concept of what civilization should be.

c. by increasing the demand for more slaves to plant and harvest more cotton

Explanation:

While the cotton gin helped in reducing the amount of work needed in removing seeds, it actually increased the need for slaves to plant and harvest the cotton and the need for land.