(i) Since \((x+1)\) and \((x+2)\) are factors of \(p(x)\), we have:

\(p(-1) = 0\) and \(p(-2) = 0\).

Substituting \(x = -1\) into \(p(x)\):

\(4(-1)^3 + a(-1)^2 + b(-1) - 2 = 0\)

\(-4 + a - b - 2 = 0\)

\(a - b = 6\) (Equation 1)

Substituting \(x = -2\) into \(p(x)\):

\(4(-2)^3 + a(-2)^2 + b(-2) - 2 = 0\)

\(-32 + 4a - 2b - 2 = 0\)

\(4a - 2b = 34\) (Equation 2)

Solving Equation 1 and Equation 2 simultaneously:

From Equation 1: \(a = b + 6\)

Substitute into Equation 2:

\(4(b + 6) - 2b = 34\)

\(4b + 24 - 2b = 34\)

\(2b = 10\)

\(b = 5\)

Substitute \(b = 5\) into \(a = b + 6\):

\(a = 11\)

(ii) With \(a = 11\) and \(b = 5\), the polynomial is:

\(p(x) = 4x^3 + 11x^2 + 5x - 2\)

We need to find the remainder when \(p(x)\) is divided by \((x^2 + 1)\).

Using polynomial division or synthetic division, the quotient is \(4x + a\) and the remainder is \(x - 13\).