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6(202^100+1)^2 + 3(202^100-2)^2 - 3(202^100+8) / 2((202^100+1)^2 - 202^100 - 202^200) (202^100-1)

First, let's simplify the expression step by step:

1. Expand the squares: = 6(408^100 + 2202^100 + 1) + 3(408^100 - 4202^100 + 4) - 3(202^100 + 8) / 2((408^100 + 2*202^100 + 1) - 202^100 - 202^200) (202^100 - 1)

2. Combine like terms: = 6408^100 + 12202^100 + 6 + 3408^100 - 12202^100 + 12 - 3202^100 - 24 / 2(408^100 + 2202^100 + 1 - 202^100 - 202^200) (202^100 - 1)

3. Simplify further: = 9*408^100 + 6 - 15 / 2(406^100 + 1 - 202^100) (202^100 - 1)

4. Combine terms: = 9*408^100 - 9 / 2(406^100 - 202^100 + 1) (202^100 - 1)

Therefore, the simplified expression is 9*408^100 - 9 / 2(406^100 - 202^100 + 1) (202^100 - 1).