Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12\) is divided by \(x ^ { 2 } + 4\).
Hence express \(\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } + 4 }\), where the values of the constants \(A , B , C\) and \(D\) are to be stated.
Use the result of part (ii) to find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x\).