Since \(p(x)\) is divisible by \((2x - 1)\), substituting \(x = \frac{1}{2}\) into \(p(x)\) gives:

\(\frac{1}{4} + \frac{a}{4} - \frac{11}{2} + b = 0\)

Multiplying through by 4 to clear fractions:

\(1 + a - 22 + 4b = 0\)

\(a + 4b = 21\)

When \(p(x)\) is divided by \((x + 1)\), the remainder is 12. Substituting \(x = -1\) into \(p(x)\) gives:

\(-2 + a + 11 + b = 12\)

\(a + b = 3\)

Solving the system of equations:

1. \(a + 4b = 21\)

2. \(a + b = 3\)

Subtract equation 2 from equation 1:

\((a + 4b) - (a + b) = 21 - 3\)

\(3b = 18\)

\(b = 6\)

Substitute \(b = 6\) into \(a + b = 3\):

\(a + 6 = 3\)

\(a = -3\)

Thus, \(a = -3\) and \(b = 6\).