(a) Substitute \(x = -1 + \sqrt{5}i\) into the equation:
\(2((-1 + \sqrt{5}i)^3) + ((-1 + \sqrt{5}i)^2) + 6(-1 + \sqrt{5}i) - 18 = 0\).
Calculate \((-1 + \sqrt{5}i)^2 = 1 - 2\sqrt{5}i - 5 = -4 - 2\sqrt{5}i\).
Calculate \((-1 + \sqrt{5}i)^3 = (-1 + \sqrt{5}i)(-4 - 2\sqrt{5}i) = 4 + 2\sqrt{5}i + 4\sqrt{5}i + 10 = 14 + 6\sqrt{5}i\).
Substitute back:
\(2(14 + 6\sqrt{5}i) + (-4 - 2\sqrt{5}i) + 6(-1 + \sqrt{5}i) - 18 = 0\).
Simplify to verify the equation holds.
(b) The conjugate root is \(-1 - \sqrt{5}i\).
Find the quadratic factor with roots \(-1 + \sqrt{5}i\) and \(-1 - \sqrt{5}i\):
\((x + 1 + \sqrt{5}i)(x + 1 - \sqrt{5}i) = x^2 + 2x + 6\).
Divide the original polynomial by \(x^2 + 2x + 6\) to find the remaining root:
\(2x^3 + x^2 + 6x - 18 = (x^2 + 2x + 6)(2x - 3)\).
The remaining root is \(x = \frac{3}{2}\).
