Internet Explorer uses Microsoft Live / Bing as its default search provider, which is nowhere near as good as Google Search. It should be pretty easy and straightforward to add Google Search and make it the default search provider: just click on a little arrow next to the search box, click on "Find More Providers..." and install your favourite search engine selecting "Make this my default search provider".
Intentionally or not, but in some instances Microsoft doesn't make it easy to find and install Google Search as your search provider.
When you click on "Find More Providers..." Internet Explorer loads a IE Gallery Search Providers page. In some localized versions of this page Google Search is simply missing from the list.
For example, if you are in the UK, Search Providers URL is http://www.iegallery.com/gb/addons/?feature=searchproviders and at the time of writing this article Google search is not there.
To find Google Search simply change gb bit in the URL to en to load a generic non localised version. Full URL would look http://www.iegallery.com/en/addons/?feature=searchproviders
