Answer: \(\frac{13}{4}-\frac{7}{4}\sqrt3\).

Work through the given conditions step by step, keeping exact values until the final answer is required.

Rewrite the expression as

\(\frac{5-\sqrt3}{(\sqrt6+\sqrt2)^2}\).

Now

\((\sqrt6+\sqrt2)^2=6+2+2\sqrt{12}=8+4\sqrt3.\)

So the expression is

\(\frac{5-\sqrt3}{8+4\sqrt3}\).

Rationalise the denominator:

\(\frac{5-\sqrt3}{8+4\sqrt3}\cdot\frac{8-4\sqrt3}{8-4\sqrt3}\).

The numerator is

\(40-20\sqrt3-8\sqrt3+12=52-28\sqrt3\).

The denominator is \(64-48=16\).

Hence the expression is

\(\frac{52-28\sqrt3}{16}=\frac{13}{4}-\frac{7}{4}\sqrt3.\)

This gives the required answer: \(\frac{13}{4}-\frac{7}{4}\sqrt3\).