Since \((x + 2)\) is a factor of \(p(x)\), we have \(p(-2) = 0\).

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

\(a(-2)^3 + 5(-2)^2 - 4(-2) + b = 0\)

\(-8a + 20 + 8 + b = 0\)

\(-8a + b + 28 = 0\)

\(-8a + b = -28\) \(\text{(Equation 1)}\)

Since the remainder when \(p(x)\) is divided by \((x + 1)\) is 2, we have \(p(-1) = 2\).

Substitute \(x = -1\) into \(p(x)\):

\(a(-1)^3 + 5(-1)^2 - 4(-1) + b = 2\)

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

\(-a + b + 9 = 2\)

\(-a + b = -7\) \(\text{(Equation 2)}\)

Subtract Equation 2 from Equation 1:

\((-8a + b) - (-a + b) = -28 - (-7)\)

\(-8a + a = -28 + 7\)

\(-7a = -21\)

\(a = 3\)

Substitute \(a = 3\) into Equation 2:

\(-3 + b = -7\)

\(b = -4\)

Thus, \(a = 3\) and \(b = -4\).