Gunboat diplomacy is most commonly associated with Theodore Roosevelt's aggressive foreign policy. Today, gunboat diplomacy is often used via naval and air deployments to troublesome areas (such as the Strait of Hormuz) to showcase the strength of the American military.

Dollar Diplomacy concerns promising loans and other lucrative economic opportunities to countries to improve diplomatic relations, most commonly associated with William Howard Taft's presidency. Taft's Dollar Diplomacy used incentives, rather than punishments to achieve its goals.

The Carrot and Stick approach involves offering a carrot (positive incentive for good behavior/improved relations), and striking with a stick (negative punishment for bad behavior). The US often used this method in Southeast Asia and the Philippines.

Woodrow Wilson is associated with "moral diplomacy," due to his passionate pursuit of humanitarian policies abroad. Woodrow Wilson accomplished much in his pursuit of a greater global good, winning over countries in agreements many foreign policy experts thought impossible.

The Monroe Doctrine was coined by President James Monroe in 1823 in an attempt to warn European nations against aggressive actions in the Western Hemisphere. The goal was to strengthen the foreign policy of the United States and to become a real world power.

The Monroe Doctrine was introduced by James Monroe in 1823, the fifth president of the United States. Theodore Roosevelt expanded upon the doctrine during his presidency, but did not introduce the original doctrine. James Munroe is also notable because he was the last Founding Father to be President.

While the Monroe Doctrine is named after James Monroe, this was simply because James Monroe was president at the time. The author of the doctrine was the Secretary of State at the time, John Quincy Adams. Monroe was the Secretary of State before Adams.

While the League of Nations was designed for many purposes, it was a dream of Woodrow Wilson for the purpose of ensuring global peace and humanitarianism. These goals were the foundation of the League of Nations, and a clear example of moral diplomacy.

American isolationism is when the United States withdraws from international conflicts, trade, and disputes, focusing all economic and political power on domestic issues like the American economy and job market, infrastructural and educational development, and social issues.

 In fewer words, isolationism is when a country limits its associations with other countries, confining itself to try and be self-reliant.

So for this question, the answer that does not demonstrate isolationism is the one that deals with America being involved in foreign affairs. The Monroe Doctrine has shaped American foreign policy a great deal, as evidenced by several Presidents invoking it. The Monroe Doctrine deals with the relationship among the countries of the Americas (North and South) and dictates that any outside nations are to stay out of the affairs among the countries of the Americas - unless they want to face a military response from the United States.

The other answers in this question all represent a form of isolationism by demonstrating a turn away from foreign affairs. The League of Nations, the precursor to the United Nations, never had U.S. participation. President Franklin D. Roosevelt wanted to expand the role of the United States on the world stage and engage more deeply in foreign affairs. Congress often rebuffed him (the extreme opposition to the U.S. joining the World Court, for example), and they stayed reluctant to engage in most foreign affairs the attack on Pearl Harbor. This is when America changed military strategy and joined in the Allies' fight in Europe and abroad.

The United States invaded Iraq in 2003 because the United States believed that Saddam Hussein had WMDs, or weapons of mass destruction (both biological and chemical). The United Nations had sent teams in to investigate and also concluded that Saddam had WMDs, with several other countries supporting this notion.  

The United States and the U.N. had worked well together in 1990 when Kuwait was invaded by Saddam Hussein and Iraq, and authorities for both the U.S. and U.N. believed the WMDs were there.  

Much of Saddam Hussein's behavior as leader of Iraq, both historically and around the time of the 2003 U.S. invasion had been suspicious, if not completely nefarious, which contributed to the U.S. and U.N.'s thought process.