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Look for a section of the website devoted to business. Usually, local governments keep official records of all businesses registered in the area, including their names. If so, there's probably a search box for you to use.  In some areas, the agency you need to check with might not be called the Secretary of State. If you're not sure, just ask a local government or development office which agency maintains business name records. For instance, in Scotland, the equivalent office is called "National Records," but in Australia it's the "Securities and Investments Commission," while in San Diego it's called the "Assessor/Recorder/County Clerk." Simply type the name into the search box and hit enter. Some government websites will also allow you to search for a specific type of business alongside the name. Leave out punctuation, plurals, and words and abbreviations like “inc.” or “co.” This help the search be as thorough as possible. For instance, if you search for “Brown bros. Painting,” the database might not reveal that there is already a company called “Brown Brothers Painting.” Search more broadly with simple terms like “Brown” and “Painting.” Record the information for any businesses listed in the database that have the name that you would like to use. If you find some with the same name, it doesn't necessarily mean that it's off-limits for you.  Sometimes, the databases include businesses that are different defunct. In such cases, you might still be able to use the name. On the other hand, even if you don't see the name that you want, that doesn't mean it's available to you. Some databases don't include businesses operating under assumed names or fictitious names. Have a list of a few possibilities so that you don't waste time scrambling to come up with a new name if a particular one is already taken. For instance, if you want to call your business "Brown Brothers Painting," you might also list some possibilities like "Brown family painting" or “Colors by Brown.”  Some business name application forms will ask you to list your first choice, followed by alternatives. Search for the alternative names as well.
Go to the website of your local Secretary of State. Enter the name you would like to search for. Make the name search as broad as possible. Make a note if the name you want turns up. Come up with some alternative names.