The problem: You have 1000 feet of fencing to build an fence surrounded by another fence with 6 feet of space between. What is the maximum area of an enclosure you can build?

Basically it is a square within a square. I have tried to assign variables with the inner polygon as L and W and the outside variables as L+6 and W+6.

Since the area is specified (though unstated in the above) to be a square, then the length is equal to the width; thus, only one variable is required. Since a lower-case "L" is often mistaken for the digit "1", I will use "w" for "width" instead, with the understanding that w = L.

However, everytime I try to solve for either L or W...

In the above, you have provided no equation. Thus, what are you "solving"?

As you know the area to be square, you therefore know the inner and outer perimeters to be 4w and 4(w + 6), respectively. Sum these, and set equal to the given total length of fencing. This is an equation in one variable. Solve for the value of the variable. Since you were given that the area is a square, you will be done at this point.