Wait on the Database Engine recovery handle failed. Check the SQL Server error log for potential causes
Following can be reason for this error

The account which user selected on Server Configuration page window ( during installation)  is somehow not able to bring SQL Server database engine services online. Either is lacks privilege or it is corrupted. During installation of database engine services SQL Server tries to bring online database services as a internal process but due to startup account either corrupt or not having appropriate privileges it fails to do so and ultimately installation fails.

Other reason is when installation fails first time due to some reason and user uninstall failed installation from add remove program, the un-installation leaves account in broken state so any further attempt to install flashes this error message.   

Solution:

Uninstall SQL Server completely from Add remove program and start new installation by right click on Setup and selecting run as admin. Now when you reach Server Configuration page please select NT Authority\System as SQL Server Database Engine service startup account.



http://social.technet.microsoft.com/wiki/contents/articles/24364.sql-server-troubleshooting-could-not-find-database-engine-startup-handle-error-during-installation.aspx

PS: If this does not helps please revert. Please follow steps mentioned in link carefully