Because they wanted to protect people's liberties. also because they wanted to ensure the rights of the people and that they were protected from the government.
Here is a list of the correct statements:
1) Ended the war between the United States and Mexico- The Treaty of Guadalupe Hidalgo ended the Mexican-American War in 1848.
2) Ceded territories from Mexico to the United States- This is also known as the Mexican Cession. The US gained territories that would later become states, such as California, Arizona, New Mexico, Utah, and Colorado.
3) Called for the US to pay Mexico $15 for lands in the Mexican Cession.
Search and seizure would mean something like the government taking your cell phone and going through every message you've ever sent, or entering your home and searching through all your dresser drawers. They need to have a court-approved reasonable cause for doing something that invasive.
Quartering of soldiers would mean soldiers would have the right to enter your home and expect you to provide them food and lodging. That's pretty invasive too.
In either case, we're talking about invasions of your privacy, your personal space. Court justices have used statements like those about search and seizure and quartering of soldiers to show that the constitution does give attention to citizen's right to privacy, even if not using the exact term "right to privacy."
Inspired ideals of life liberty and the pursuit of happiness as well as separation of powers. John Locke and Baron de Montesquieu
