How to enable REGEX for RC6 SQLite?

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Might be, but wouldn't have anything to do with OP's question.
OP's question is about using the REGEXP-Operator/Function within SQL-Statements.
And in DB Browser's case i'd guess they compiled the extension and linked it statically into DB Browser itself (not into sqlite3.dll!), but "load" it into the engine on startup

I seem to remember that RC6 could use VB6 functions as custom functions for sqlite.