Algebra issues.... please help me. i don't get the basics?

Is that whole thing on the left supposed to be in the numerator? Is it:

[ (z^2 + z + 1)(1) - (z+1)(2z+1) ] / (z^2 + z + 1) ?

If that's the case, then just focus your attention on the numerator. Multiplying the first term doesn't change anything, so this is:

[ z^2 + z + 1 - (z+1)(2z+1) ] / (z^2 + z + 1)

Now we have to expand the (z+1)(2z+1). Use F.O.I.L. or the distributive rule twice:

[ z^2 + z + 1 - (z(2z+1) + 1(2z+1)) ] / (z^2 + z + 1)

[ z^2 + z + 1 - (2z^2 + z + 2z+1) ] / (z^2 + z + 1)

[ z^2 + z + 1 - (2z^2 + 3z+1) ] / (z^2 + z + 1)

Now distribute the minus sign and combine like terms. The +1 and -1 cancel each other out. The z^2 and -2z^2 become -z^2. And the z with the -3z become -2z:

[ z^2 + z + 1 - 2z^2 - 3z - 1 ] / (z^2 + z + 1)

[ z^2 + z - 2z^2 - 3z ] / (z^2 + z + 1)

[ -z^2 + z - 3z ] / (z^2 + z + 1)

[ -z^2 - 2z ] / (z^2 + z + 1)