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?

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?

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 model and year are your Macs?

Well, let's not look a 🎁🐴 in the 😦.