Hello!  Normally Microsoft Office 2010 came on a set of discs that would have the product key on them.  Or it would be on an office 2010 card that should have been kept in a safe place.  As far as I've ever been able to tell, Dell does not keep any product keys on file for their users.  

Thanks - seems stupid regarding maintaining inventory control. You would think for a tech company it shouldn't be that difficult

MPI cards were essentially sealed and just placed in the systems box. As they were sealed and the key was hidden... Dell has no record of the individual product key. 

Microsoft may help you with the product activation if you provide them your systems original configuration but its doubtful given its Office 2010 and they are trying to enforce Office 2016 subscriptions.

You may be able to recover your key but this depends on how you purchased Office 2010.

For example,

If you purchased the box product, the key would be in the packaging. If you lost the packaging there may still be a way to get a new product key.

If you purchased it online through Digital River or the Microsoft Store, you should be able to recover the product key online.

If the key was provided by your school or company’s IT department, you will likely need to contact them about a replacement.

factory installed Microsoft Office 2010 has a Microsoft Product ID card with the Key.

https://www.dell.com/support/article/en-us/sln115308/how-to-locate-the-microsoft-product-identifier-card-for-your-microsoft-office-2010-purchase?lang=en