Verified answer

Too much working. Let a, b, c and d be roots.

Given a = 2i, means b=-2i

abcd = 24, and ab = 4, so cd = 6

a+b+c+d = 1, and a+b = 0, so c+d=1

So c = 1/2 + ig and d = 1/2 - ig.

cd = 1/4 + g² = 6, so g² = 23/4, so g = √23/2

and hence the roots are 2i, -2i, 1/2 + i√23/2, and 1/2 - i√23/2.

if 2i is a root then -2i is also a root

(x - 2i)(x + 2i) = x^2 + 4 is a factor of the polynomial

divide to get another quadratic: x^2 - x + 6 = 0

Use the quadratic formula: x = [1 +- sqrt(1 - 4(1)(6))]/2 = [1 +- sqrt(-23)]/2

x = 1/2 +- sqrt(23)i

and x = 2i, - 2i