(i) To find the value of \(a\), we divide \(p(x) = x^4 + 3x^3 + ax + 3\) by \(x^2 - x + 1\). Performing polynomial division, we obtain a quotient of \(x^2 + 4x + 3\) and a remainder of zero, which confirms divisibility. Solving for \(a\), we find \(a = 1\).

(ii) With \(a = 1\), the polynomial becomes \(x^4 + 3x^3 + x + 3\). Factoring or solving this equation, we find the real roots are \(x = -3\) and \(x = -1\).