How would I solve this question involving surds

Question: A rectangular box has a square base of side (3 + 2√5) cm. The volume of the box is (153 + 80√5) cubic centimeters. Find the height of the box, giving your answer in surd form.
if possible, could you please explain how you got the answer along with the working? thank you so much.
Bonus question: in what part of mathematics would you classify surds under?(this is just a random question)

Area of square base = (3 + 2√5)(3 + 2√5)

Area of square base = 9 + 6√5 + 6√5 + 4√5√5

Area of square base = 9 + 12√5 + 4(5)

Area of square base = 29 + 12√5

Height of box = Volume / Area of base

Height of box = (153 + 80√5) / (29 + 12√5)

Now we have to rationalise the denominator by multiplying the top and bottom lines by the same thing, namely (29 - 12√5)

Height of box = [(153 + 80√5)(29 - 12√5)] / [(29 + 12√5)(29 - 12√5)]

Height of box = (4437 - 1836√5 + 2320) / (841 - 348√5 + 348√5 - 144(5))

Height of box = (6757 - 1836√5) / (841 -720)

Height of box = (6757 -1836√5) / 121 cm

Please check my arithmetic.

When confronted with something of the form (a + b√c) / (d + e√f) always rationalise by multiplying the top and bottom by (d - e√f). This will have the effect of removing all surds from the bottom line.

I would place surds in algebra.