Scaling up a graphic that has a 300 pixels/inch resolution?

Here's a handy little formula for ppi (note ppi not dpi).

ppi required = 6878/viewing distance in inches.   Anything above that is wasted as normal good eyes will not resolve it.

It comes from this:

A good human eye can resolve 60 line pairs per degree i.e. pairs of black and white lines.  So in half a degree we would have 60 single lines.

Simple trigonometry means that the width containing those 60 lines can be calculated by:
               Width of 60 lines = Viewing distance x tan(0.5°)

Once we know the width for 60 lines (or pixels) we can calculate how many lines per inch can be seen by using 60/Width of 60 lines.

So putting those together gives us

60/(Viewing distance x tan(0.5°)) = 60/(Viewing distance x 0.0087269)

= 2/(Viewing distance x 0.000291)    

Or

1/(Viewing distance x .0001454)

Or

6878/Viewing distance

Dave

Unless you're lowering the DPI, doubling the image size is going to cause some pixellization.  I tend to do smaller images at higher DPIs (i.e. 600-800 dpi) so if I need to blow it up for print, I can, but keep it at the standard 200-300 DPI for print.

It's not a standard, defacto or otherwise. It is simply the right value for some output. 

Yes, I suppose one could say in general, 'most' output to halftone work used a 150LPI with 2X quality factor. As the LPI went up, 2X was overkill, and 1.5 was more than enough. Output ranged from very low (double digits for newspapers) to above 200LPI and more. Start talking photo printers, totally different. My old Kodak XL7700 was 203 DPI, that's what you sent to it (203PPI), it was true contone output. Film recorders; are different. Inkjet, different. One of the better pieces on what to select is Jeff Schewe's from many years back:
https://www.digitalphotopro.com/technique/photography-workflow/the-right-resolution/

When someone actually specifies what kind of printing is being done, then we can get closer to an optimal recommended PPI for the image. The size of course matters as does the viewing distance.

The audience too as the late great Bruce Fraser would say: "The distance a photographer views a print is based upon the length of his nose." 

No joke, at a gallery opening for Mac Holbert of Nash Editions and Dano Steinhart of Epson, this did happened:

I wish the urban legend that 300 PPI (DPI) is a standard for print would die. It isn't. Any more than 72P PPI is for displays.

In the old days when halftone printing at a common but not the only LPI was 150 and many used a quality factor of 2X, providing 300 PPI was often used. As often as the line screen wasn't 150 and the QF wasn't 2X. In those days no one would send 300 PPI for newspaper printing or dye subs like my XL7700, two of many examples of printing from digital images.

Since you are in the ballpark, but just a bit under optimum, I suggest you upsample to 150% (using "Preserve Details"). This will actually get you more pleasing edges (not as much "oversharpening" artifacts; i.e. the light halos around objects) than if you upsampled to 200%, which is what we usually tend to do.

Your first challenge is to maintain quality on enlargement. Although 300 ppi is a standard for print, it is not always required; it depends on both the printing method and the viewing distance. For example, if you are printing on inexpensive matte paper or a color laser printer, 200–250 ppi might be enough because the printing process might not resolve more than that. For viewing distance, a 150 ppi print looks as good as a 300 ppi print if the 150 ppi print will be normally viewed from twice as far away as the 300 ppi print.

If you want to maintain 300 ppi at 8 inches tall, you can enlarge using Image > Image Size with the Resample option enabled; that might look good enough although not perfect. The best results for a resampled enlargement might come from using advanced AI-powered enlargement software that tries to enhance detail, such as the Super Resolution feature in Adobe Lightroom Classic/Camera Raw, the Super Zoom Neural Filter in Photoshop, or Topaz Gigapixel AI.

Your second challenge is that you can’t scale 5.35 × 3.55 inches to 10 × 8 inches, because they are different proportions. If 10 × 8 inches is required, you have to resolve the difference by either fitting the entire image within 10 × 8 inches (leaving empty areas across the top and bottom), or cropping off the ends (if that’s OK). If you must show the entire image uncropped and 8 inches tall, the print must be 12 × 8 inches.

Without to change image at all, actually image dimension in pixels you can change, lower print resolution as already explained. Otherwise go and upscale image with some loss of quality which may not be that drastical and maybe acceoptable. Duplicate image then upscale and take a look, perhaps it is ok for you because upscaling up to 100% sometimes works just fine.

180-200PPI is about the minimum I'd consider for printed output on a photo printer (depending too on size and viewing distance). 

Working with one dimension: 

IF you're saying you have 5.35x300 pixels, that 1605 total pixels on that one axis. At 180PPI, that's 8.9 inches so a tad shy. 

What you can do is use the Image Size dialog, set it so it does NOT interpolate then type the size (let's say 10 inches) you desire and let the value fall where it is. (that's only 160PPI), and from there, you might consider resampling to 180 (1800 pixels on that long axis). There is no free lunch here! You'll lose some quality compared to having enough pixels from the beginning but you're not that far off. And if you can live with a 9 inch print, you're pretty much there.