Since \(p(x)\) is divisible by \((2x - 1)\), we have \(p\left(\frac{1}{2}\right) = 0\).

Substitute \(x = \frac{1}{2}\) into \(p(x)\):

\(a\left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^2 + b\left(\frac{1}{2}\right) + 3 = 0\)

\(\frac{1}{8}a + \frac{1}{4} + \frac{1}{2}b + 3 = 0\)

Multiply through by 8 to clear fractions:

\(a + 2 + 4b + 24 = 0\)

\(a + 4b = -26\)

When \(p(x)\) is divided by \((x + 2)\), the remainder is 5, so \(p(-2) = 5\).

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

\(a(-2)^3 + (-2)^2 + b(-2) + 3 = 5\)

\(-8a + 4 - 2b + 3 = 5\)

\(-8a - 2b = -2\)

\(8a + 2b = 2\)

Now solve the system of equations:

1. \(a + 4b = -26\)

2. \(8a + 2b = 2\)

Multiply equation 1 by 2:

\(2a + 8b = -52\)

Subtract equation 2 from this result:

\(2a + 8b - 8a - 2b = -52 - 2\)

\(-6a + 6b = -54\)

\(-a + b = -9\)

\(b = a - 9\)

Substitute \(b = a - 9\) into equation 1:

\(a + 4(a - 9) = -26\)

\(a + 4a - 36 = -26\)

\(5a = 10\)

\(a = 2\)

Substitute \(a = 2\) back to find \(b\):

\(b = 2 - 9 = -7\)

Thus, \(a = 2\) and \(b = -7\).