Using only the digits 0,1,2,3, write down all 
possible three digit numbers with sum of digits 3,
allowing numbers to begin with 0, in order from 
largest to smallest:

300
210
201
120
111
102
030
021
012
003

in each, use the digits of each 3-digit number 
in the order they appear as exponents of 
a, b, and c respectively, as this scheme shows:

300 --> a3b0c0 = a³
210 --> a2b1c0 = a²b
201 --> a2b0c1 = a²c
120 --> a1b2c0 = ab²
111 --> a1b1c1 = abc
102 --> a1b0c2 = ac²
030 --> a0b3c0 = b³
021 --> a0b2c1 = b²c
012 --> a0b1c2 = bc²
003 --> a0b0c3 = c³

Rewrite the dividend a³+b³+c³-3abc putting in zero
place-holders for each of those not represented in
that dividend, in that order.  That is, the dividend 
becomes:

a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³

So we start with this:
         _______________________________________________________________ 
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³

Then we divide a³ by a, getting a², and we write that as the 1st term of the quotient.

          a²                                                            
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
         

Then we multiply that by each term of the divisor and place each product
under the term that it is like, and draw a line under it:

          a²                                                            
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
          a³ +  a²b +  a²c

Then we subtract and bring EVERY term down

          a²                                                            
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
          a³ +  a²b +  a²c
               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³

Then we divide -a²b by a, getting -ab, and we write that as the 2nd term of the quotient:

          a² - ab                                                       
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
          a³ +  a²b +  a²c
               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³

Then we multiply that by each term of the divisor and place each product
under the term that it is like, and draw a line under it, subtract and bring
EVERY term down:

          a² - ab                                                       
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
          a³ +  a²b +  a²c
               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
               -a²b        -  ab² -  abc
                      -a²c +  ab² - 2abc + 0ac² +  b³ + 0b²c + 0bc² + c³

Keep doing that, and end up with this:

          a² - ab - ac + b² - bc  + c²                                  
a + b + c)a³ + 0a²b + 0a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
          a³ +  a²b +  a²c
               -a²b -  a²c + 0ab² - 3abc + 0ac² +  b³ + 0b²c + 0bc² + c³
               -a²b        -  ab² -  abc
                      -a²c +  ab² - 2abc + 0ac² +  b³ + 0b²c + 0bc² + c³
                      -a²c        -  abc -  ac²
                              ab² -  abc +  ac² +  b³ + 0b²c + 0bc² + c³
                              ab²               +  b³ +  b²c
                                  -  abc +  ac² - 0b³ -  b²c + 0bc² + c³
                                  -  abc              -  b²c -  bc²
                                            ac² - 0b³ + 0b²c +  bc² + c³
                                            ac²              +  bc² + c³
                                                                       0


So the answer is: a² - ab - ac + b² - bc  + c²

Edwin