A cuboid has base area \(\left(4+2\sqrt{5}\right)\ \mathrm{cm}^2\) and a volume of \(\left(9+5\sqrt{5}\right)\ \mathrm{cm}^3\). Find, without using a calculator, the height of the cuboid, in cm, giving your answer in the form \(\dfrac{1}{2}\left(a+b\sqrt{5}\right)\), where \(a\) and \(b\) are integers.[3]
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