Chris:  Thank you for contacting the UVM Extension Master Gardener Helpline with your question on how to control Goutweed. Aegopodium podagraria (also known as Bishop's Weed or Goutweed) is listed on the Vermont Noxious Weed Watch List, because of its ability to spread quickly by underground rhizomes.

There are a few ways you can try to control this weed.  The method you use depends on whether you have other plants located in the same area that you may not want to remove.  If the goutweed is located in an area with other plants, the best way is to pull the goutweed.  However, the root system is extensive, so when pulling the plants, the roots and rhizomes also need to be pulled out.  Vermont Invasives recommends that after pulling, bag the plant material and leave for a week before throwing the bag in the trash.  DO NOT COMPOST, or toss in the woods, or other areas.

For a large area (or an area where you want to eliminate everything growing in that area), you can smother the weed out.   First, mow the area very close, then lay black plastic over the entire area and seal all of the edges with ground staples, or boards, etc.   This area should be kept covered for about a year.   If this is an area where you don't want to leave the plastic exposed, it can be covered with mulch to hide the plastic.

Another option is to use an herbicide that is specifically recommended for goutweed.  The attached link (Gallery of Terrestrial Plants) discusses which type of herbicide to use that will reach the roots.  Otherwise, the top of the plant will die, but the roots are still alive and will sprout new growth.  Again, if your infestation of goutweed is in an area with other plants or trees, you may not to use this method.

This link will take you to the Vermont Invasives web site page that lists plants on the prohibited and watch lists.  Scroll down to Goutweed and click for more information.

Gallery of Terrestrial Plants | Vermont Invasives (vtinvasives.org)

Below is a fact sheet from Vermont Invasives:

goutweed2010.pub (vtinvasives.org)

Other misc. information from the University of Maine that you may find interesting.

Weed Management Post-Webinar Q&A | Extension (unh.edu)

How do you get rid of Bishop's Weed? - Cooperative Extension: Garden and Yard - University of Maine Cooperative Extension (umaine.edu)

I hope you find this information helpful.   These control methods may need to be repeated for a year or two, but eventually should help the situation.  Good luck in your efforts!