When you search from Cortana or Start Menu in Windows 10, Web Search generated Bing urls will be opened in default browser with Bing, here Microsoft doesn’t allow the default search engine in browser to handle them, Mozilla wants the user to have a choice, that’s why they want Windows 10 passed off web Searches to open with the default search provider in default browser.
READ: Windows 10 Now lets you Turn off ‘Web Results’ in Search
Firefox to use default Search engine for Windows 10 Web Search
For this, they’ve added and enabled ‘Use this search engine for searches from Windows’ option in ‘Search’ category of preferences below the ‘Provide Search Suggestions’. This feature has been targeted for Firefox 42 version.
If you use Firefox as default browser (and set Google as default Search engine) in Windows 10 current build 10240, ensure you’ve the latest version of Nightly, and click on Start and type a search query and click on first web result.
ALSO SEE: Make Web Search in Windows 10 to use Google instead of Bing
Then Search results in Firefox will be opened with Google (since you’ve set Search engine giant as default) instead of Bing.