EXHIBIT 3.251

State of Delaware

Secretary of State

Division of Corporations

Delivered 03:51 PM 12/20/2006

FILED 03:51 PM 12/20/2006

SRV 061170153 — 4272332 FILE

CERTIFICATE OF FORMATION

OF

DESERT HOSPITAL HOLDINGS, LLC

Under Section 18-201 of the

Delaware Limited Liability Company Act

FIRST: The name of the limited liability company is Desert Hospital Holdings, LLC (the “Company”).

SECOND: The address of the registered office of the Company in the State of Delaware is 2711 Centerville Road, Suite 400, Wilmington, Delaware 19808, County of New Castle.

THIRD: The name and address of the registered agent for service process on the Company in the State of Delaware is Corporation Service Company, 2711 Centerville Road, Suite 400, Wilmington, Delaware 19808, County of New Castle.

IN WITNESS WHEREOF, the undersigned has executed this Certificate of Formation as of December 20, 2006.

By: /s/ Rebecca Hurley

Name: Rebecca Hurley

Title: Authorized Person