How to Remove the "File" and "File Collection" from the Search Menu?

You'll need to implement hook_form_FORM_ID_alter in a new custom module, then alter the "type" field options. Alter form ID search_block_form for the header search tool, and search_form for the advanced search.

Note that this will not actually remove the media and media_collection node types from the search index, only from the UI. For v7.2, we are considering adding an admin option to include/exclude types via the search index config screen as well.

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