Rút gọn

To simplify the expression, we can start by multiplying the numerator and denominator of each fraction separately.

For the first fraction, we have:
(1 + 2 + √2) / (1 + √2)

For the second fraction, we have:
(1 - 2 - √2) / (1 - √2)

Now, let's simplify each fraction individually.

For the first fraction:
(1 + 2 + √2) / (1 + √2)
= (3 + √2) / (1 + √2)

For the second fraction:
(1 - 2 - √2) / (1 - √2)
= (-1 - √2) / (1 - √2)

Now, let's multiply the two fractions together:
((3 + √2) / (1 + √2)) * ((-1 - √2) / (1 - √2))

To simplify this expression, we can use the difference of squares formula, which states that (a^2 - b^2) = (a + b)(a - b).

Applying this formula, we have:
((3 + √2) / (1 + √2)) * ((-1 - √2) / (1 - √2))
= ((3 + √2) * (-1 - √2)) / ((1 + √2) * (1 - √2))
= (-3 - √2 - 3√2 - 2) / (1 - 2)
= (-5 - 4√2) / (-1)
= 5 + 4√2

Therefore, the simplified expression is 5 + 4√2.