A garden fence with a gate

A community garden is rectangular, with one side along a straight wall. No fence is needed along the wall. The other three sides are enclosed using all of \(40\text{ m}\) of fencing and a separate gate \(2\text{ m}\) wide. The gate is fitted in the side opposite the wall and does not use any of the fencing.

Each side perpendicular to the wall has length \(x\) metres. The full length of the side parallel to the wall, including the gate, is \(L\) metres, where \(L>2\). The wall is long enough for all dimensions considered.

1. Show that the area \(A\text{ m}^2\) of the garden is given by \(A=42x-2x^2\).[3]
2. Express \(A\) in completed-square form.[3]
3. Hence find the greatest possible area of the garden and the corresponding values of \(x\) and \(L\).[2]
4. Find both possible pairs of dimensions when the area of the garden is \(216\text{ m}^2\).[3]