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

1. Set up the expression for the height of the cuboid

$\text{Height} = \frac{\text{Volume}}{\text{Base Area}} = \frac{9 + 5\sqrt{5}}{4 + 2\sqrt{5}}$

2. Rationalise the denominator by multiplying by the conjugate

$\text{Height} = \frac{9 + 5\sqrt{5}}{4 + 2\sqrt{5}} \times \frac{4 - 2\sqrt{5}}{4 - 2\sqrt{5}}$

3. Expand and simplify the numerator and denominator

$\text{Numerator} = (9)(4) + (9)(-2\sqrt{5}) + (5\sqrt{5})(4) + (5\sqrt{5})(-2\sqrt{5})$
$= 36 - 18\sqrt{5} + 20\sqrt{5} - 50$
$= -14 + 2\sqrt{5}$

$\text{Denominator} = 4^2 - (2\sqrt{5})^2$
$= 16 - 20$
$= -4$

4. Simplify the fraction to the required form

$\text{Height} = \frac{-14 + 2\sqrt{5}}{-4}$
$= \frac{-2(7 - \sqrt{5})}{-4}$
$= \frac{7 - \sqrt{5}}{2}$
$= \frac{1}{2}(7 - \sqrt{5})$