\nWe need to simplify the expression $\\frac{4}{1+\\sqrt{2}+\\sqrt{3}}$. This involves rationalizing the denominator, which contains multiple surds.
\nWe can group the terms in the denominator, for instance, as $(1+\\sqrt{2}) + \\sqrt{3}$. We multiply the numerator and denominator by the conjugate of this expression, which is $(1+\\sqrt{2}) – \\sqrt{3}$.
\nExpression = $\\frac{4}{(1+\\sqrt{2})+\\sqrt{3}} \\times \\frac{(1+\\sqrt{2})-\\sqrt{3}}{(1+\\sqrt{2})-\\sqrt{3}}$
\nNumerator = $4(1+\\sqrt{2}-\\sqrt{3})$.
\nDenominator = $((1+\\sqrt{2})+\\sqrt{3})((1+\\sqrt{2})-\\sqrt{3})$. This is in the form $(a+b)(a-b) = a^2 – b^2$, where $a = (1+\\sqrt{2})$ and $b = \\sqrt{3}$.
\nDenominator = $(1+\\sqrt{2})^2 – (\\sqrt{3})^2$.
\n$(1+\\sqrt{2})^2 = 1^2 + 2(1)(\\sqrt{2}) + (\\sqrt{2})^2 = 1 + 2\\sqrt{2} + 2 = 3 + 2\\sqrt{2}$.
\n$(\\sqrt{3})^2 = 3$.
\nDenominator = $(3 + 2\\sqrt{2}) – 3 = 2\\sqrt{2}$.
\nSo the expression becomes $\\frac{4(1+\\sqrt{2}-\\sqrt{3})}{2\\sqrt{2}}$.
\nWe can simplify by dividing 4 by 2: $\\frac{2(1+\\sqrt{2}-\\sqrt{3})}{\\sqrt{2}}$.
\nNow, we need to rationalize the denominator $\\sqrt{2}$ by multiplying the numerator and denominator by $\\frac{\\sqrt{2}}{\\sqrt{2}}$.
\nExpression = $\\frac{2(1+\\sqrt{2}-\\sqrt{3})}{\\sqrt{2}} \\times \\frac{\\sqrt{2}}{\\sqrt{2}}$
\nExpression = $\\frac{2\\sqrt{2}(1+\\sqrt{2}-\\sqrt{3})}{2}$.
\nCancel the 2 in the numerator and denominator:
\nExpression = $\\sqrt{2}(1+\\sqrt{2}-\\sqrt{3})$.
\nNow, distribute $\\sqrt{2}$:
\nExpression = $\\sqrt{2} \\times 1 + \\sqrt{2} \\times \\sqrt{2} – \\sqrt{2} \\times \\sqrt{3}$
\nExpression = $\\sqrt{2} + 2 – \\sqrt{6}$.
\nRearranging the terms to match the options, we get $2 + \\sqrt{2} – \\sqrt{6}$, which is the same as $\\sqrt{2}-\\sqrt{6}+2$.
\n<\/section>\n