Urgent: 4 degree polynomial with irrational roots

Hi

which specific formula/method I can use to find the complex roots of a polynomial of degree 4 on form:

which has irrational coefficients

a = and b = and

where

If I write the equation as , then you will notice that it factorises as

Then by fundamental theorem of algebra I see that

where the complex roots of the original polymial p(x) is

Getting back to the original polynomial

(*)

where

I would like to prove that a complex number x makes (*) true iff

is a root of the

I see that that

So the solutions of in my equation are .

I then need to plug each of these numbers into the equation , which can be written as . Using the quadratic formula again, you get .

By the way do x^-1 then exist ?

Yes, there's no problem about that. In fact since , you can see that

I get this x value for both values of s to be:

This expression makes the equation

true

since

This proves that there exist a number x which is both a root of the original polynomial and Q(s = x + x^{-1}), since .

But one question remains on my part. Since x is supposedly a complex number. Is it a complex number in its current form? That I'm a bit unsure of. Since then it should be written in the form x = a + bi ?

Best Regards,

Billy