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I would like to search on Google a keyword "key1" together with "key2" or "key3".

Would it work "key1" and (key2 or key3) or "key1" and key2 or "key1" or key3? Which way would be better to search a keyword (that must be present) among with others (any of them would work).

For instance if they keyword "dog" must be be present and, in addition, also one of the following should be included: cat, food, trick, pet.

What would be the best way to build the search query? Might "dog" and (cat or food or trick or pet) work?

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Google's default search operator is AND. Google also ignores punctuation unless it's part of a search operator, and parentheses are not part of a search operator.

Further, OR is given higher precedence in the order of operation than AND.

So, in your last example, you'd want

"dog" cat OR food OR trick OR pet  

You can also go to https://www.google.com/advanced_search and construct your query that way, and then observe how it puts the search operators together.

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