Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(2 ( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
Hence write down the minimum value of \(2 x ^ { 2 } + 6 x + 5\).
The point \(A\) has coordinates \(( - 3,5 )\) and the point \(B\) has coordinates \(( x , 3 x + 9 )\).
Show that \(A B ^ { 2 } = 5 \left( 2 x ^ { 2 } + 6 x + 5 \right)\).
Use your result from part (a)(ii) to find the minimum value of the length \(A B\) as \(x\) varies, giving your answer in the form \(\frac { 1 } { 2 } \sqrt { n }\), where \(n\) is an integer.