The search page required is a single page form with multiple free-form and drop down search selections. How to go about creating custom search page in SharePoint 2010?
Build a custom web part in Visual Studio that performs a search, you can have as many controls as you want. This would require an understanding of building custom web parts, which can't be imparted in a forum such as this.
If you have a programming background in VB.NET or C#, you should be able to pick up the differences. You can google "Building Custom Web Parts in SharePoint 2010", here is a fairly decent one I found:
Check this post may it help you to customize the advanced search out of the box web part
All of the above are correct answers. There are many solutions to this problem. I would also suggest using the SharePoint Search Application's asmx web services, with the community-standard jQuery library SPServices to call ws.
This is will result in a light-weight sandbox-able ajax search web part.