(i) Since \((x + 2)\) is a factor of \(p(x)\), substituting \(x = -2\) into \(p(x)\) should yield zero:

\((-2)^3 - 2(-2) + a = 0\)

\(-8 + 4 + a = 0\)

\(a = 4\)

(ii) With \(a = 4\), the polynomial becomes \(x^3 - 2x + 4\). To find the quadratic factor, divide \(x^3 - 2x + 4\) by \(x + 2\):

Perform polynomial division:

\(x^3 - 2x + 4 = (x + 2)(x^2 - 2x + 2)\)

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