When will I stop taking problems from creators, again?

I don't create this problem. All of the credits go to Nguyễn Tất Thu, Đoàn Quốc Việt and Vũ Công Minh for writing this problem, and the book in general.

Here is a slightly altered version of the original problem:

Solve the following equation:

$\large x^2 + \left(\dfrac{12\sqrt 5}{5} - 2\right)x - \sqrt{-x^2 + 4x + 5} + \dfrac{36 - 9\sqrt 5}{5} = 0$

If $x = \dfrac{a - b\sqrt 5}{5}$ , calculate the value of $a - 5b^2$ .