A cuboid has a square base of length $(3 + \sqrt{2})$ cm and a height of $h$ cm. The volume of the cuboid is $(19 - 3\sqrt{2})$ cm$^3$. Without using a calculator, obtain an expression for $h$ in the form $(a + b\sqrt{2})$, where $a$ and $b$ are integers.

1. Set up the equation for the volume of the cuboid

Volume = Base Area $\times$ Height
$(3 + \sqrt{2})^2 \times h = 19 - 3\sqrt{2}$

2. Expand the base area

$(3 + \sqrt{2})^2 = 3^2 + 2(3)(\sqrt{2}) + (\sqrt{2})^2$
$= 9 + 6\sqrt{2} + 2$
$= 11 + 6\sqrt{2}$

3. Express $h$ as a fraction

$h = \frac{19 - 3\sqrt{2}}{11 + 6\sqrt{2}}$

4. Rationalise the denominator to find $h$

$h = \frac{19 - 3\sqrt{2}}{11 + 6\sqrt{2}} \times \frac{11 - 6\sqrt{2}}{11 - 6\sqrt{2}}$
$= \frac{19(11) - 19(6\sqrt{2}) - 3\sqrt{2}(11) + 3\sqrt{2}(6\sqrt{2})}{11^2 - (6\sqrt{2})^2}$
$= \frac{209 - 114\sqrt{2} - 33\sqrt{2} + 18(2)}{121 - 36(2)}$
$= \frac{209 - 147\sqrt{2} + 36}{121 - 72}$
$= \frac{245 - 147\sqrt{2}}{49}$
$= 5 - 3\sqrt{2}$