Help with expanding and simplifying surds?

Always remember this: (x+y)^2 = x^2 + 2xy + y^2
In your case, x = √6, y=2√3.

x^2 = (√6)^2 = 6
2xy = 2(√6)(2√3) = 4√18
y^2 = (2√3)^2 = 4(3) = 12

Adding all three terms:
= 6 + 4√18 + 12
=18+4√18

( √6 + 2√3 ) ( √6 + 2√3 )

6 + 4 √18 + 12

18 + 12 √ 2

(√6 + 2√3)^2
= (√6 + 2√3)(√6 + 2√3)
= √(6 * 6) + 2√(3 * 6) + 2√(6 * 3) + (2 * 2)√(3 * 3)
= 6 + 2√18 + 2√18 + 4(3)
= 6 + 4√18 + 12
= 18 + 4√18

The first answer in wrong, the second is correct (18+4root(18)). Sometimes solutions manuals can be wrong.