The statement that best explains the scandals associated with the Harding administration is:
"Harding's cabinet used their appointment as a way to get rich any way they could."
This statement captures the essence of the corruption that characterized Harding's presidency, including the infamous Teapot Dome scandal, where cabinet members engaged in bribery and unethical practices for personal gain. While there were other issues, this statement broadly encompasses the general atmosphere of corruption within the administration.