In the top part of the image, I illustrate synthetic division. I get the numbers by writing the -2 on the left since we are dividing by x+2. Then I write down the coefficients of the polynomial. Next I bring down the first coefficient and then I mulitiply it by the -2 and add it to the next coefficient and write it down. I repeat the process until I get to the end of the line. So the next 1 came from -2(1)+3. Then the next 1 comes from -2(1)+3 again. Lastly the -4 comes from -2(1)+-2.
I would like to eventually make a little interactive program to go through the steps of synthetic division and long division by hand. If you subscribe to my RSS feed at http://www.lsquaredmath.us you can be notified when I make new programs.
For the long division problem, you start off with the x+2√(x3+3x2+3x-2) and then you multiply by x2 first since x×x2 matches the first term of the polynomial p(x). Then you multiply the x2 by the x+2 to get the x3+2x2. Subtract to get the next line and repeat the process until you get to the end.
Note the coefficients of the answers to both methods match.