(i) To verify that \(1 + 2i\) is a root, substitute \(x = 1 + 2i\) into the equation \(2x^3 + x^2 + 25 = 0\).

Calculate \((1 + 2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4 = -3 + 4i\).

Calculate \((1 + 2i)^3 = (1 + 2i)(-3 + 4i) = -3 + 4i - 6i + 8i^2 = -3 - 2i - 8 = -11 - 2i\).

Substitute into the equation: \(2(-11 - 2i) + (-3 + 4i) + 25 = -22 - 4i - 3 + 4i + 25 = 0\).

Thus, \(1 + 2i\) is verified as a root.

(ii) Since the coefficients of the polynomial are real, the complex roots occur in conjugate pairs. Therefore, the other complex root is \(1 - 2i\).

(iii) On the Argand diagram, plot the point \(1 + 2i\). The equation \(|z| = |z - 1 - 2i|\) represents the perpendicular bisector of the line segment joining the origin \(O\) and the point \(A = 1 + 2i\). This is a straight line passing through the midpoint of \(OA\) and intersecting \(OA\) at right angles.