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While the Book Review plugin does support adding custom links to your reviews (including Amazon), the upcoming Affiliate Linkalizer for Amazon plugin will make that process much easier by automatically constructing the URL for your Amazon buy links.
The first thing you'll want to check is that you have created a Google API Key and that it has been entered correctly on the Advanced tab of the Book Review Settings.
This message can also appear if the IP address of your web server has not been entered correctly in the Google Developers Console. You will need to go back into the Developers Console and enter the correct IP address. If you're still having trouble, try removing the IP address altogether. See the documentation for more information.
If all else fails, please leave a message in the General Support forum. Be sure to include the URL of your web site in your post.
The Synopsis field is meant to hold the summary or description of the book. It is not where you are intended to write your review. You should write your review in the regular WordPress editor at the top of the page, although if you prefer you can certainly write it inside the Synopsis editor instead. Be aware that if you do that, the custom links will show at the very bottom of your post, rather than directly below the book's description and cover image.
Titles that start with "A", "An" or "The" are sorted using the second word in the title. For example, The Hunger Games would be shown as Hunger Games, The. If you have a post that is not sorting this way, try going back into that post and re-saving it.
Check to ensure that you have filled out the Genre field in the Book Info section of the post, and that the Include post in archives checkbox is selected.
If you are showing thumbnails of the book covers in your archives, then you should be aware that thumbnails are only used in those themes that support Featured Images. If your theme does not support Featured Images, or if a post does not have a Featured Image set, then scaled-down versions of the full-size covers are used. The images are scaled down as per the Thumbnail size value in the Media Settings. Showing thumbnails in the archive(s) without using Featured Images will result in longer page load times.