(i) Since \((2x - 1)\) is a factor of \(p(x)\), substituting \(x = \frac{1}{2}\) into \(p(x)\) gives:

\(\frac{1}{8}a + \frac{1}{4}b + \frac{5}{2} - 2 = 0\)

\(\Rightarrow \frac{1}{8}a + \frac{1}{4}b + \frac{1}{2} = 0\)

\(\Rightarrow a + 2b + 4 = 0\) (Equation 1)

When \(p(x)\) is divided by \((x - 2)\), the remainder is 12. Substituting \(x = 2\) into \(p(x)\) gives:

\(8a + 4b + 10 - 2 = 12\)

\(\Rightarrow 8a + 4b + 8 = 12\)

\(\Rightarrow 8a + 4b = 4\) (Equation 2)

Solving Equations 1 and 2 simultaneously:

From Equation 1: \(a = -2b - 4\)

Substitute into Equation 2:

\(8(-2b - 4) + 4b = 4\)

\(-16b - 32 + 4b = 4\)

\(-12b = 36\)

\(b = -3\)

Substitute \(b = -3\) into Equation 1:

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

\(a - 6 + 4 = 0\)

\(a = 2\)

(ii) With \(a = 2\) and \(b = -3\), the polynomial is \(2x^3 - 3x^2 + 5x - 2\).

Divide \(2x^3 - 3x^2 + 5x - 2\) by \(2x - 1\) to find the quadratic factor:

The division gives a quotient of \(x^2 - x + 2\).

Thus, the quadratic factor is \(x^2 - x + 2\).