If the area of a rectangle is A, why is the area of a rectangle with lengths twice as long not 2A?

This is because you are doubling both the length of the rectangle and its width. If it were extended by a factor of 2 in only one direction then its are would be 2A. Extending it in the other direction as well gives dimentions of 22A=4A. Generally, when a shape with area A has its directions increased by a factor of n then the resultant area of the shape is nnA or (n^2)A