To simplify the given expressions, we will simplify each term separately and then combine them.

1. Simplifying the first expression:
- First, let's simplify the terms inside the square root:
- √6 + √5
- Since these are irrational numbers, we cannot simplify them further.
- Now, let's square the sum of the square roots:
- (√6 + √5)^2 = (√6)^2 + 2(√6)(√5) + (√5)^2 = 6 + 2√30 + 5 = 11 + 2√30
- Next, let's simplify the term outside the square root:
- 1/2√120 = 1/2√(2^2 * 2 * 3 * 5) = 1/2 * 2√(2 * 3 * 5) = √30
- Finally, let's simplify the last term:
- √(15/12) = √(15/4) = √(3 * 5/2^2) = √(3 * 5)/2 = √15/2

Putting it all together:
1/2(√6 + √5)^2 - 1/2√120 - √(15/12) = 1/2(11 + 2√30) - √30 - √15/2 = 11/2 + √30 - √30 - √15/2 = 11/2 - √15/2

2. Simplifying the second expression:
- First, let's simplify the terms inside the square root:
- √20 = √(2^2 * 5) = 2√5
- √45 = √(3^2 * 5) = 3√5
- √125 = √(5^3) = 5√5
- Now, let's simplify the terms outside the square root:
- 1/2√20 = 1/2 * 2√5 = √5
- 1/3√45 = 1/3 * 3√5 = √5
- Finally, let's simplify the last term:
- √125 = 5√5

Putting it all together:
1/2√20 + 1/3√45 - √125 = √5 + √5 - 5√5 = √5 - 4√5 = -3√5

Therefore, the simplified expressions are:
1. 11/2 - √15/2
2. -3√5