You wrote: Could we represent the length of a segment with a "letter" and use the calculator to make all the necessary calculation? (I know this is not possible YET, but can it be included in a future version???) We could then represent n^2 like a real square, and dynamically varying (wash, my english is so bad) the length of the segment "n" to see the square "expanded". The software would then represent the area of the square like n^2 and NOT with the "real" number. We could then represent thing like 2n^2 + n + 3 "geometrically" but still have an unknow to deal with.
You can certainly construct a segment and label it "n".
You can then construct a square of side n, area = n^2.
Then you can drag to vary n to see the change in the square and value of n^2.
To obtain 2n^2 + n + 3, you need to decide what 1 is. This is determined by the unit chosed in preferences, or it can be defined. You can help this further by using axes and a grid. I like to set the preferences at cm. Then I place a point on the screen, then translate it 1 cm. Now I construct a 1 cm segment, then choose it at the unit for the graph. With the grid on the screen, everything is 1 cm. Now 2n^2 + n + 3 can be represented geometrically, just as you could represent it with algebra tiles.
What do you mean by "solving" the equation. Do you mean where does 2n^2 + n + 3 = 0? You can actually graph the function y = 2n^2 + n + 2 and see where it crosses the x-axis. If you want to do symbol manipulation, then use a symbol manipulator like Mathematica, Math Exploration Toolkit, Derive, etc. Geometer's Sketchpad and graphing calculators provide a visual representation of the relationship -- which seems to be easier to understand than just moving symbols around.