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External drives are a great idea for backup. That is how they should be used. If you need another bootable device, it would be a good idea to try a bootable USB flash drive instead of using your external hard drive.
Of course, you can backup your operating system to the external drive. And then you would need to have a suitable boot DVD that would put up a recovery program that would read from your external hard drive and restore your system. But didn't your Seagate external drive come with software that does an?
As a general rule, the Microsoft operating system is not licensed for use on removable media. However, Microsoft does allow a special boot CD called a PE disc. Would you like to know more about that?