Since \(x^2 + x + 2\) is a factor of \(p(x) = x^4 + 3x^2 + a\), we can perform polynomial division of \(p(x)\) by \(x^2 + x + 2\).

1. Divide \(x^4 + 3x^2 + a\) by \(x^2 + x + 2\) to get a quotient of \(x^2 - x + 2\).

2. The remainder must be zero for \(x^2 + x + 2\) to be a factor.

3. Equate the constant remainder to zero and solve for \(a\):

\(a = 4\).

Thus, the other quadratic factor is \(x^2 - x + 2\).