4 / (1 + √2 + √3) is equal to

To simplify the expression 4 / (1 + √2 + √3), we rationalize the denominator. We group the terms as (1 + √2) and √3, then multiply both the numerator and denominator by the conjugate, (1 + √2 - √3).

Step 1: Simplify the Denominator
((1 + √2) + √3)((1 + √2) - √3) = (1 + √2)² - (√3)²
= (1 + 2 + 2√2) - 3 = 2√2

Step 2: Simplify the Expression
4(1 + √2 - √3) / 2√2 = 2(1 + √2 - √3) / √2

Step 3: Final Division
Distributing the division by √2:
(2 / √2) + (2√2 / √2) - (2√3 / √2) = √2 + 2 - √6

Rearranging the terms gives √2 - √6 + 2, which corresponds to option D.