### Working with complex symbolic expressions

I am a beginner at Sage so my questions may not be well informed, so bear with me.

The context: I am trying to use sage to explore the exponential function assuming all I know about it is that it is its own derivative and has the value 1 at z = 0. It is then easy to develop the Taylor series to any degree using formula like:

expp2( z ) = 1 + ( 1/ factorial( 1 ) )* z + ( 1/ factorial( 2 ) ) * ( z ^ 2 )

You can take 2 of these for z = a and z = b and multiply them together in Sage. Getting something like:

1/4*(b^2 + 2*b + 2)*(a^2 + 2*a + 2)

Now I have done the algebra by hand and know this reduces to the Taylor expansion for z = a + b.

What I do not get is how to show this part in Sage in a nice clean way ( I have some ways I do not like so much ).

Here I have 2 questions one specific, and one general ( I am interested in the answer to either one or both):

1) If this exponential question interests anyone, could you offer some tips? I have tried various ( but not all ) applications of expand and simplify. I am still plugging away.

2) Is there a guide that would help me learn how to carry out algebraic operations over complex expressions. I have looked at several basic tutorials, but they do not have much detail. The most useful single resource I have found is

http://www.sagemath.org/doc/reference/sage/symbolic/expression.html

and the pages linking from it.