Planting under a tree is problematic - the competition with roots and moisture makes it difficult for plants to get a foothold. There are some perennials that will grow in dry shade, like barrenwort. Offhand, I can't think of any more. I assume the roots you mention are from the tree?

Yes it is from the tree, but the area was able to handle weeds and forget me nots and money plants (luminara annua) grow pretty well there. A rose of sharon is not more than 6 feet from the tree and it does quite well.

The short answer to your situation is raised beds. They are an excellent solution to soil that can't be made friendly to growing plants and give you the opportunity to have a fertile, almost ideal growing medium for whatever you decide to plant.

Then you need to consider the shade/sun situation you have. Is the 4 hours in the heat of the day or early morning/late afternoon? A lot of shade lovers can take some sun if it's not too intense.

Vegetables or ornamentals is the next consideration. Few vegetables will be ideal with that much shade but root crops and leafy crops do reasonably well with part sun if it's in the more intense times of day and the shade is light from high limbed trees. Tomatoes and other "fruiting" veggies require more sun to do well. They tend to get very tall and produce lots of vegetative growth with little energy going to making the parts we want to eat when they get less than 6 hours of good sun a day.

Some things to think about.

Sandy

Here's the deal. My plans call for a circular garden (because the patio encircles it) where the above ground pool was. About 2 inches down you hit pine roots. I can cut out the roots and amend the soil for my current plan which is a cross pattern of stone with a centerpiece of perhaps a statue or something and then plants in each of the four corners. I dread to think of how I could possible cut a raised bed to fit inside those pie pieces. Plus, wood tends to rot under ground in my soil (had a raised bed that fell apart about a year ago)

In regards to the sun, the sun hits the area from about 9-10 AM and then from about 3-6:30ish PM. Other than that its mostly in shade with dappled light on occasion