Express \(x ^ { 2 } + 3 x + 2\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
A curve has equation \(y = x ^ { 2 } + 3 x + 2\).
Use the result from part (a) to write down the coordinates of the vertex of the curve.
State the equation of the line of symmetry of the curve.
The curve with equation \(y = x ^ { 2 } + 3 x + 2\) is translated by the vector \(\left[ \begin{array} { l } 2 4 \end{array} \right]\). Find the equation of the resulting curve in the form \(y = x ^ { 2 } + b x + c\).