We start by dividing \(p(x) = x^4 + 3x^3 + ax + b\) by \(x^2 + x - 1\). The division gives a quotient of the form \(x^2 + kx\) and a remainder \((a+3)x + (b-1)\).

Since the remainder is given as \(2x + 3\), we equate the coefficients:

\(a + 3 = 2\) and \(b - 1 = 3\).

Solving these equations gives:

\(a + 3 = 2 \Rightarrow a = -1\)

\(b - 1 = 3 \Rightarrow b = 4\)

Thus, the values are \(a = -1\) and \(b = 4\).