What did president james polk do to gain additional u. s. territory, including california and new mexico?

I guess It depends upon if you are a Mexican or an American. 
a Mexican might say C. Polk picked the war in order to annex the land. It was a war of native aggression over land, sold to Americans at the time as defending US honor. 

An American might say B. Yeah we went to war and invaded, and had troops garisoning Mexico City.. but they signed a treaty. 
Kind of signed a treaty at the end of a gun barrel though. But Mexico did sign the treaty. 

The entire answer is C, then B. and we wouldn't have gotten B, without C. Polk believed in Manifest Destiny.. That it was America's pre ordained destiny to expand to the Pacific Ocean. As such he picked a war with Mexico in order to take the land necessary for America to achieve this goal. 

D is just wrong... Nueces River was Mexican recognized pre war boarder of Texas.. the post war boarder was the Rio Grande. 

A is wrong too.. Austin died 1836 before polk was in office and before the US annexed Texas in Dec 1845. 

First Polk got a declaration of war, kinda tried to make it look like Mexico was the aggressor.. but we know today it was Polks plan all along to provoke the war and blame it on Mexico.. So Polk invaded Mexico.. and finally with US troops occupying Mexico City he got Mexico to sign the Treaty of Guadalupe Hidalgo... 

In this treaty the US annexed 1.2 million square miles of territory to the United States; Mexico's size was halved, while that of the United States increased by a third. California, Nevada, Utah, most of Arizona, and parts of New Mexico, Colorado and Wyoming were all included in the Mexican Cession. The treaty also recognized the annexation of Texas and acknowledged American control over the disputed territory between the Nueces River and the Rio Grande. Mexico, in turn received $15 million dollars.

Hope I helped!

James Polk instigated war with Mexico, which resulted in the Mexican-American War. As a result, we gained Texas, California, and New Mexico.