A few things. Some grasses spread, some don't. What kind of grass do you have? If you don't know, then where do you live? Fertilizer can increase spread rate, but this is usually at the detriment of your lawns health, and the rate of spread won't be dramatically faster. Your dogs are a big part of your problem. I love dogs, and I love a good lawn, but if they are very active and big, your lawn will suffer. Overseeding is always an option, but you will have to keeps the dogs off the lawn for at least 8 weeks, even then they will probably hurt the new grass.

If you just pretend you have grass seed out and start watering daily, 3x per day, for about 10 minutes; then you will have a lawn full of crabgrass by July.

Have you considered putting down mulch instead? You can usually get several yards of chipped tree mulch for free from a tree trimmer with a tow behind chipper/shredder. Dogs love the stuff. Make it as deep as you can - 18 inches is real good and will put a stop to the mud.

Just noticed that the area I listed is wrong. It is only 40 X 40. It is in front of our great room window. So, wood chips would work. I think I will just over seed and water like crazy! Thx. to all.