How do you know if a goat has passed her after birth?
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The dried blood is normal. If she had a retained afterbirth she would be probably showing signs of infection by now. Such as an elevated temperature, droopy ears, smelly discharge, etc.
If she retained a placenta she would eventually start acting ill. If she does start acting off, take her temperature, since fever would be a strong indicator that an infection was beginning. Cassie covered what to look for!
Goats, like humans, experience what is called lochia after giving birth. It is a discharge that can last several weeks. It is more red for a few days after birth, then slowly turns brown, eventually to whiteish, before ceasing. If you see large quantifies or bright bright red blood, that is a problem. But a little bit of red ooze is just the lochia.
It gets very crusty around the vulva and tail. I usually use clippers to clear away excess hair around a doe's vulva before kidding time, to keep the mess under control. I use a warm wet cloth to remove crust once or twice a day, swiping with a new part of the cloth from cleanest to dirtiest in the area (dirtiest would be their bum, so start at lady bits so as not to contaminate).
Lochia is basically the expulsion of all sorts of things like fetal membranes and uterine lining that built up during pregnancy. Other stuff too, but that is what it is in a nutshell.
You can try pulling it. Use a slow steady traction. Don't yank or pull too hard it will break off. If she is still open you can insert a couple uterine boluses made for cows. They help break the retained afterbirth down and fight infection. If you can't get your hand in there you can't use them.