Since \((3x + 1)\) is a factor of \(p(x)\), substituting \(x = -\frac{1}{3}\) into \(p(x)\) should yield zero:

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

\(-\frac{1}{27}a + \frac{1}{9}b - \frac{1}{3} + 3 = 0\)

\(-\frac{1}{27}a + \frac{1}{9}b + \frac{8}{3} = 0\)

Multiply through by 27 to clear fractions:

\(-a + 3b + 72 = 0\)

\(a = 3b + 72\) (Equation 1)

When \(p(x)\) is divided by \((x - 2)\), the remainder is 21. Substitute \(x = 2\) into \(p(x)\):

\(p(2) = a(2)^3 + b(2)^2 + 2 + 3 = 21\)

\(8a + 4b + 5 = 21\)

\(8a + 4b = 16\)

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

Substitute \(a = 3b + 72\) from Equation 1 into Equation 2:

\(2(3b + 72) + b = 4\)

\(6b + 144 + b = 4\)

\(7b + 144 = 4\)

\(7b = -140\)

\(b = -20\)

Substitute \(b = -20\) back into Equation 1:

\(a = 3(-20) + 72\)

\(a = -60 + 72\)

\(a = 12\)

Thus, the values are \(a = 12\) and \(b = -20\).