To simplify the expression, we first need to find a common denominator for all three terms.
The first term has a denominator of 1 + a + a^2, the second term has a denominator of 1 - a + a^2, and the third term has a denominator of 1 + a^2 + a^4.
To find a common denominator, we need to factor each of the three denominators.
The first denominator can be factored as (a + 1/2)^2 + 3/4.
The second denominator can be factored as (a - 1/2)^2 + 3/4.
The third denominator can be factored as (a^2 - a√2 + 1)(a^2 + a√2 + 1).
Now we can rewrite the expression with a common denominator:
[(a + 2)((a - 1/2)^2 + 3/4) - (a - 2)((a + 1/2)^2 + 3/4) - 2a^2(a^2 - a√2 + 1)(a^2 + a√2 + 1)] / [(a + 1/2)^2 + 3/4][(a - 1/2)^2 + 3/4](a^2 - a√2 + 1)(a^2 + a√2 + 1)]
We can simplify this expression by multiplying out the terms in the numerator and combining like terms. After doing so, we get:
[-4a^4 + 8a^3 - 4a^2 - 4a√2 + 12a - 4√2] / [(a + 1/2)^2 + 3/4][(a - 1/2)^2 + 3/4](a^2 - a√2 + 1)(a^2 + a√2 + 1)]
So the simplified expression is:
(-4a^4 + 8a^3 - 4a^2 - 4a√2 + 12a - 4√2) / [(a + 1/2)^2 + 3/4][(a - 1/2)^2 + 3/4](a^2 - a√2 + 1)(a^2 + a√2 + 1