Question
I live in Clearwater, Fl and have St. Augustine grass. First, my neighbors' St. Augustine front yard is now overgrown with Bermuda that is now entering my yard...yikes! He also cuts his grass so low...I would think that by cutting mine at 3 inches would protect mine, but it seems not. Secondly, my backyard has a flat, broadleaf weed that spreads in all directions, looks like a flat fern, and is profusive, encroaching and destroying what was once a beautiful backyard. I have had chemical treatments for the last three years to no avail. The county extension thought it could be basket or Jew grass, but on looking it up on the internet, it has no resemblence! No one locally can identify it, it seems. Sidebar: I was cutting my neighbor's yard recently, who once had a nice yard, then I noticed this same weed spreading all over his yard with little St. Augustine remaining. I have dogs and am concerned about the chem treatments, as well. I have a  partially shaded backyard, soil recently treated with lime due to low pH. I really want my backyard to a place of peace and harmony...my respite, if you will. There are squirrels, birds, butterflies, black snakes (yikes, again),opossums and rabbits on occasion... I fear my only remedy, which is costly, is to tear up the yard and start over, however, the roots to the weeds are very, very deep. Please advise. Thank you for your time and help. Blessings! Nancy

Answer
Ok Nancy- let me see if I am getting your concerns right,
#1- Your front yard is being invaded by your neighbors Bermuda grass.
         The only thing you can do here is to put up some kind of a barrier between your yard and his(I'm assuming there is nothing there now.) Unfortunately since you are taking good care of your part of the yard whatever he has in his yard will want to spread into yours where the conditions are more favorable.

#2-The weeds in your back yard.
      Please please please stay away from chemicals! When you use chemicals your lawn almost becomes addicted to them like drugs. What they do is they replace all the good things that nature supplies to your yard and when the chemicals run out there is nothing there to support your lawn, so it gets yellow and ugly until you either replace the chemicals or nature eventually starts to return after all of the chemicals have been washed away.
      What type of lawnmower do you use? Is it a mulching lawnmower? It sounds like it is because of how you explain the weeds spreading. Some weeds are spread because when they are cut up,the pieces will grow right where they land. With these kind of weeds it makes a nightmare for you.
      Right before you cut the grass take about 15 min or so and get rid of the weeds that you can, by pulling them out.I know that you said they are very very deep,but this is the best thing to do.Grab the weed as close to the ground as you can and give it a good slow pull so you will get the root.If you water really well the day before it will help with this, the soil will be moist so the weeds will come out easier. You don't have to do them all at the same time, just do a little at a time every time you mow. The more you let it all grow the easier it will be to spot them, and with the regular watering, your soil will be nice and moist so it will be so easier to get them by the roots.
                Eventually you will have a weed free, healthy green lawn that was even better than before, but it will take time, patience and some work. If you are dedicated to it you will get results and you will actually be doing less work because nature will be doing all the work for you. Before you know it all you'll have to be doing is pulling the odd weed mowing, and watering.
        I think this should get you started, and if you have anymore questions or just need clarification or whatever, you know what to do!