Q.Loosestrife gooseneck perennial (white)
I planted 2 plants about five years ago and they have taken over my garden! The roots spread out in a complex network and even encroach on my lawn and have invaded my Hosta. How do I get rid of it?
Thank you.

Go the the garden store and get an herbicide that has the chemical, glyphosate, in it (such as 'Roundup'). I've found it cheaper to buy the glyphosate as a concentrate so that I can mix a lot of it at home in my two-gallon hand sprayer. Pick a warm, windless, and rain-free day and very carefully but thoroughly spray the gooseneck loosestrife with the glyphosate. Be very careful not to spray any plants that you want to keep. Glyphosate will kill just about any plant it touches if enough of it gets on them. However, with gooseneck loosestrife, you may have to make several applications throughout the growing season. Wait at least a week before applying glyphosate again if you didn't get all of it the first time.

You will need to use a Glyphosate herbicide to kill Loosestrife.
It is however, non-selective and will kill other plants that it touches.
Careful application of the herbicide is very important to protect your other plants.
https://extension.psu.edu/natural-resources/water/news/2013/gooseneck-loosestrife-an-arch-beauty-quick-to-escape
https://www.gardeningknowhow.com/ornamental/flowers/gooseneck-loosestrife/gooseneck-loosestrife-flowers.htm