Rationalizing a denominator. The question instructs to rationalize the denominator in the following fraction:

My solution is as follows:

The book's solution is 

which is exactly the numerator in my solution.
Can someone confirm my solution or point what what I'm doing wrong?
Thank you.

Recall when we have a product that expands to a "difference of squares", we obtain just that: a difference that we obtain, not a sum:
$$(a - b)(a+b) = a^2 - b^2$$ So your denominator should expand to $$(\sqrt 6 - 2)(\sqrt 6 + 2) = (\sqrt 6)^2 - 2^2 = 6 - 4 = 2.$$
Correcting for this gives us the correct answer: $$\dfrac{10 + 4\sqrt 6}{2} = 5 + 2\sqrt 6.$$
