Unfortunately, at the size of your screenshot there could be a million "artifacts" and none would be discernible. Here's what I suspect you're seeing (because I can't): the text from the screenshot becomes pixelated doggy-doo when you print.
To get clean-looking text on a desktop inkjet printer, you need real text (vector data), or your document as a whole must be a minimum 300 ppi or higher at the printed size. Images are much more forgiving, but text is not. Our eyes expect text to be pin sharp, so anything less is visible and offensive.
In this case, you have three problems combining to produce visual sewage:
Your document is only 996 pixels wide. (Ignore the "72 ppi" bit; "per inch" is only there to tell software what size -- pixels divided by "pixels per inch" -- to print it. You can make the "per inch" anything you like; the image doesn't change.) Your print software has only those pixels to work with, so at 8 inches wide you have an effective resolution of about 125 ppi.
All text that is part of the original screenshot is already rasterized and has at least some anti-aliasing applied.
Photoshop applies its own anti-aliasing to text added with the text tool, depending on the setting you chose in the control panel and the absolute resolution of the document. The bad news is that Photoshop doesn't send any vector data to the printer, so that pixelated, anti-aliased image of the text is what actually prints. That would be okay if all those pixels weren't then being enlarged, but they are.
Whereas Illustrator and other vector programs "think" in Postscript or a close relative, Photoshop does not. Vectors, to Photoshop, are just a handy way of corralling pixels. So when you ask your printer driver (probably not the most sophisticated image processor out there) to scale up this image, it is scaling not just the text itself, but also the anti-aliasing pixels added by Photoshop.
The net result of that is an attack of the uglies.
So, to answer your question, there is no way to print this screenshot as-is without serious degradation of the text. What you could do, if you felt it worth the trouble, would be to scale it up in Photoshop to 8.5 inches wide at 300 ppi, then remove all the text in the screenshot and replace it with new text typed in Photoshop. You will still have some degradation in the image, but your text will look reasonably crisp.
Otherwise, convert the document resolution to 300 ppi and print without scaling. That will give you the best result, and plenty of extra space on the paper for notes. :-)