Previously, you could look on the Utah Corporation Commission website and see the public record corporate filings for Utah corporations. However, now they charge $2 per copy to see the PDF filings. I assume that applies to your own corporation in Utah. Since I am too cheap to pay the fee, I guess I can't show the recordings.
But before Utah began imposing the fee, I found out that legally, the name "FamilySearch" is a registered tradename of the Genealogical Society of Utah, which is in turn a DBA or "doing business as." Here is the explanation from the FamilySearch.org Research Wiki:
The Genealogical Society of Utah (GSU) is dedicated to gathering, preserving, and sharing genealogical information throughout the world.The corporate entity of FamilySearch is known as FamilySearch International and does business as Genealogical Society of Utah and FamilySearch. The Genealogical Society of Utah is listed as a former business name. See FamilySearch International.
Established in 1894, the GSU is an incorporated, nonprofit educational institution entirely funded by The Church of Jesus Christ of Latter-day Saints. Its headquarters are in Salt Lake City, Utah, USA with local representatives in all parts of the world.
In the 1990's, The Church of Jesus Christ of Latter-day Saints expanded the Genealogical Society of Utah and it eventually became known as FamilySearch. During the 1990's, Monte J. Brough, a General Authority and Executive Director of the Family History Department and president of the Genealogical Society of Utah, conceived of an Internet genealogy service which he proposed to church leaders. His ideas eventually developed into what today is known as FamilySearch.org. In 1999, the new website, FamilySearch.org, went live, providing an online medium for making genealogical records easily available to the public anywhere in the world at no cost.
Not too long ago, the Genealogical Society of Utah had a separate website. However, it appears that the site has been taken down. The Encyclopedia of Mormonism has this explanation of the history:
The Genealogical Society of Utah, organized in 1894, became The Genealogical Society of The Church of Jesus Christ of Latter-day Saints in 1944. In 1976 it became The Genealogical Department, and in 1987 the name was changed to The Family History Department. Each name change brought renewed emphasis and expanded resources to further the search for ancestors. The name Genealogical Society still continues as the microfilm section of the Family History Department of the Church.If you want a very detailed history of the entire organization and the process of how the Genealogical Society of Utah became FamilySearch, see the book
The central purpose of the organization is expressed in a statement by Elder Joseph Fielding Smith: "Salvation for the dead is the system whereunder those who would have accepted the gospel in this life, had they been permitted to hear it, will have the chance to accept it in the spirit world, and will then be entitled to all the blessings which passed them by in mortality" (DS 2:100-196). Provisions have been made, therefore, for the living to provide, vicariously, ordinances of salvation for their deceased family forebears and friends. This cannot be done without information about the dead.
Allen, James B., Jessie L. Embry, and Kahlile B. Mehr. Hearts Turned to the Fathers: A History of the Genealogical Society of Utah, 1894-1994. Provo, Utah: BYU Studies, Brigham Young University, 1995.