Proof that polynomial multiplication works I would like to understand why polynomial multiplication works the way it is defined.  For example, we know that $(x+1)^2 = x^2+2x+1$, but how can we prove that this actually works?  
More generally, how did we came up with the method used to multiply polynomials?
Any insight would be appreciated.
 A: In addition to crasic's symbolic answer, one can also look at the geometric interpretation of multiplication of two quantities (the area of the rectangle formed with sides of the given length) and find that it is consistent.

$(x+a) * (x+b) = x^2 + ax + bx + ab$
Edit: As pointed out by Douglas, the geometric interpretation is only defined at all when $x$, $a$, and $b$ are positive, so care must be taken to keep algebraic self-consistency when generalizing to the real numbers.  Geometric "proofs" should only be used as an "intuition check", in any case.
A: If you have a set $X$ and two real valued functions $f$, $g:X\to\mathbb R$, then there is a standard way to multiply $f$ and $g$: the product is the function $h:X\to\mathbb R$ such that $$h(p)=f(p)g(p)$$ for all elements $p$ of $X$.
If $X=\mathbb R$ and if $f$ and $g$ are polynomials, then the function $h$ turns out also to be a polynomial, given by the usual formula. This completely determines the multiplication rule.
A: We define the property that multiplication has to distribute over addition. 
That is $A(B+C) = AB + AC$
With polynomials we try to do something similiar, analogous to the real numbers we all know so well.
$(x+1)^2 = (x+1)(x+1) = (x+1)x + (x+1)1 = (x^2 + x) + (x + 1) = x^2 + 2x + 1$
Hence, by the distributive law of multiplication you can see how the defined operation makes sense. 
You CAN come up with some binary operation that combines polynomials and doesnt distribute over addition, however this operation would not be called multiplication.
