Answer: \(14-\sqrt{10}\).

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

First combine the \(+1\) with the fraction:

\(\frac{16+11\sqrt{10}}{2+\sqrt{10}}+1=\frac{16+11\sqrt{10}+2+\sqrt{10}}{2+\sqrt{10}}=\frac{18+12\sqrt{10}}{2+\sqrt{10}}.\)

Rationalise the denominator:

\(\frac{18+12\sqrt{10}}{2+\sqrt{10}}\cdot\frac{2-\sqrt{10}}{2-\sqrt{10}}\).

The numerator is

\(36-18\sqrt{10}+24\sqrt{10}-120=-84+6\sqrt{10}\).

The denominator is \(4-10=-6\).

So the expression is

\(\frac{-84+6\sqrt{10}}{-6}=14-\sqrt{10}\).

This gives the required answer: \(14-\sqrt{10}\).