(i) Since \((x - 2)\) is a factor of \(p(x)\), by the factor theorem, \(p(2) = 0\).

Substitute \(x = 2\) into \(p(x)\):

\(p(2) = 2^3 - 3a(2) + 4a = 0\)

\(8 - 6a + 4a = 0\)

\(8 - 2a = 0\)

\(2a = 8\)

\(a = 4\)

(ii)(a) Substitute \(a = 4\) into \(p(x)\):

\(p(x) = x^3 - 12x + 16\)

Divide \(p(x)\) by \((x-2)\) to find the other factor:

\(x^3 - 12x + 16 = (x-2)(x^2 + 2x - 8)\)

Factor \(x^2 + 2x - 8\):

\(x^2 + 2x - 8 = (x-2)(x+4)\)

Thus, \(p(x) = (x-2)^2(x+4)\)

(ii)(b) Substitute \(x^2\) into \(p(x)\):

\(p(x^2) = (x^2 - 2)^2(x^2 + 4)\)

Set \(p(x^2) = 0\):

\((x^2 - 2)^2 = 0\) gives \(x^2 = 2\) so \(x = \pm \sqrt{2}\)

\((x^2 + 4) = 0\) gives \(x^2 = -4\) so \(x = \pm 2i\)

The roots are \(\pm \sqrt{2}, \pm 2i\).