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

\(-\frac{6}{8} + \frac{1}{4}a - \frac{1}{2}b - 2 = 0\)

Simplifying, we get:

\(-\frac{3}{4} + \frac{1}{4}a - \frac{1}{2}b - 2 = 0\)

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

Multiplying through by 4 gives:

\(a - 2b = 11\) (Equation 1)

Since the remainder when \(p(x)\) is divided by \((x + 2)\) is \(-24\), substituting \(x = -2\) into \(p(x)\) gives:

\(-48 + 4a - 2b - 2 = -24\)

Simplifying, we get:

\(-50 + 4a - 2b = -24\)

\(4a - 2b = 26\)

Dividing through by 2 gives:

\(2a - b = 13\) (Equation 2)

Solving Equations 1 and 2 simultaneously:

From Equation 1: \(a = 11 + 2b\)

Substitute into Equation 2:

\(2(11 + 2b) - b = 13\)

\(22 + 4b - b = 13\)

\(3b = -9\)

\(b = -3\)

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

\(a - 2(-3) = 11\)

\(a + 6 = 11\)

\(a = 5\)

Thus, \(a = 5\) and \(b = -3\).