How do I keep my Mom from feeding scraps to the dog?

If it is her dog, you can't. If it is your dog, you keep him next to you at the table.

Is the dog begging from the table, or is mom doing it, after the meal? If it is the first, I would remove the dog from the dining room.

Since your mother has dementia, it is not likely that reasoning with her or "teaching" her or making her promise or explaining the reasons is going to help. You'll have to be diligent to keep the dog away from Mother when Mother has food.

djm, it is not unusual for households to feed scraps to their pets. If this was commonly done in your Mother's household, then it will be difficult to break the habit. And with your Mother having memory issues, you may not be able to stop her from doing that.

The best thing to do is shadow your mother when there is food on the table... either keep the dog out of dining area, or if you see your mother putting food scraps into the dog bowl quickly remove the scraps if the dog allows you... it might not be easy.