(1.00001) ^ 2 * (0.99999) ^ 2 - (1.00002) ^ 2 * (0.99998) ^ 2

This can be re-written as,
(1.00001 * 0.99999)^2 - (1.00002 * 0.99998)^2 (I'm too lazy to multiply this So, multiply here)

Now the formula a^2-b^2 = (a+b)(a-b).

With the help of answer choices, you do not need to multiply these numbers and instead go by approximations.

What are the answer choices?

I think it's ridiculuos to calculate the solution? Is there a shorter method via estimation?

Think of x as 10^-5

We have (1+x)^2 * (1-x)^2 - (1+2x)^2 * (1-2x)^2

This is (1-x^2)^2 - (1 - 4x^2)^2

which in turn in (2 - 5x^2)(3x^2) = 6x^2 - 15x^4

i.e. 6*10^-10 - 15*10^-20

0.00000000059999999985