if a book says that 4+3=5 or 3+2=5 then it is not relevant which book it is

Um... yes, but I didn't ask what book you were using. I asked how -5x

^{2} + 4x + 6 related to x

^{4} - 5x

^{2} + 6, and what solution method you were using.

How does solving this quadratic equation help you find the roots of the original quartic equation? How do these two different equations with different solutions relate? What method are you using that applies the Quadratic Formula to this different equation in order to find solutions to the original equation?

the book says that the 4 roots are sqrt(2), sqrt(-2), sqrt(3), sqrt(-3)

is this correct or not

The answer hasn't changed from the other replies: Yes, the book's answer is correct.

is there a method you can use to arrive at the book answers

Yes. The method I outlined in my first reply, which I pointed out again in a later reply and another poster applied for you, will lead to the book's answers.

(That's assuming, of course, that you are not familiar with radicals, so you don't know that

and

are two very different things. The latter is the correct form.)