This exercise asks you to get access to a computer algebra system. I suggest to use SINGULAR, but if you prefer to use one that you already know (e.g. Macaulay2 or MAGMA), you can use that instead.
The homepage for SINGULAR is here. You should get access to it in one of the following ways:
bramble.stanford.edu. At the command line type
~galatius/Singular/Singular(to run it in a terminal) or
~galatius/Singular/ESingular(to run it inside Emacs).
ring r = 0,(x,y,z),dp;
poly f = x^3 + y^3 + z^3;
poly g = (x+y+z)*(x^2 + y^2 + z^2);
f; f+g; f*g;
ideal I = f,g;