my Mac will not recognize my external drive and thus will not back up

Ok. What model is your MacBook Pro? If it is an Intel-based Mac, I suggest that you reset the SMC on it to see if resolves why it is not appearing in the Disk Utility.

Ref: How to reset the SMC on your Mac - Apple Support

If this doesn't help, then the issue could be the drive itself, or its cable. Have you verified that it can be used successfully with another device?

I still will need to know the answers to my other questions: drive format & does it appear in the Disk Utility. Also, if it does appear in the utility, please provide a screen shot of the utility's screen.

What is the make & model of this external drive? What is it formatted as: HFS+, APFS, NTFS, etc.? Does this drive appear in the Disk Utility?

What model and year are your Macs?

Well, let's not look a 🎁🐴 in the 😦.