Final answer:
Compensated emancipation was not a specific legislative effort proposed for the issue of slavery in the territories during the Mexican-American War period. Instead, the era focused on popular sovereignty, the Wilmot Proviso, and the Kansas-Nebraska Act.
Step-by-step explanation:
The solutions proposed for the issue of slavery in the territories around the time of the Mexican-American War included various legislative efforts, but not all of them specifically dealt with the compensation for the emancipation of enslaved people. The options provided refer to: 1) Compensated emancipation, which is not historically linked to the specific proposals in question; 2) Popular sovereignty, the idea that the settlers of a territory would vote to decide whether slavery would be legal; 3) The Wilmot Proviso, a legislative proposal that aimed to ban slavery in any territory acquired from Mexico; and 4) The Kansas-Nebraska Act, which allowed the territories of Kansas and Nebraska to decide for themselves whether to allow slavery, in effect nullifying the Missouri Compromise.