I lost my Sims 3 pets key code to install the game

If your Sims 3 and Origin accounts are connected, then the game should appear in your My Games on the Origin client. If it does, then right click on the game and click - View Game Details. On the right hand side, you will see "product code" and the code underneath it.

If the game does not appear in your My Games, you can also find it in the registry.

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If you still have the game on your computer you can follow these steps to find your serial code from your registry (PC only). Credits of to Carl's Sims 3 Guide

You need to be signed in as an Administrator first. Windows may require your permission to open the
Registry Editor and if so, just click on yes.

Press the Windows key on your keyboard and type regedit in the search bar and press Enter.
Click on 'File' at the top left of the window that opens then click on Export.
Choose a location and name for the file as this will be your backup registry file.
Click Save.
This backup is created just as a precaution.
Now to find your serial codes for each specific EP or SP installed on your computer:

Expand the HKEY_LOCAL_MACHINE folder.
Expand the SOFTWARE folder (Wow6432Node for 64 bit systems)
Expand the Electronic Arts folder.
Expand the Sims folder
Expand the EP or SP you need the code for
Click on ergc and in the box to the right under 'Data' you will see the code
Write it down somewhere safe
Repeat above steps for each EP
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Otherwise, you will need to contact a game advisor: http://eahelp.ea.com/en/ . For help with navigating the site, see this: http://answers.ea.com/t5/Beta-Archive/How-to-contact-a-Game-Advisor-using-the-new-EA-Help-page/m-p/2161281

For more information: http://answers.ea.com/t5/The-Sims-3/FAQ-Lost-and-broken-DVDs-and-missing-serial-codes-What-to-do/td-p/14230

Best of luck and happy gaming