Let's solve the given problems step-by-step.

Problem 20:

Simplify $5\sqrt{3} - 2\sqrt{12}$.

1. Simplify $\sqrt{12}$: $\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}.$

   So, the expression becomes: $5\sqrt{3} - 2(2\sqrt{3}) = 5\sqrt{3} - 4\sqrt{3}.$

2. Combine like terms: $5\sqrt{3} - 4\sqrt{3} = \sqrt{3}.$

Final Answer for 20: $\sqrt{3}$.

Problem 22:

Simplify $(6 + \sqrt{2})(3 - \sqrt{2})$.

1. Use the distributive property (FOIL): $(6 + \sqrt{2})(3 - \sqrt{2}) = (6 \cdot 3) + (6 \cdot -\sqrt{2}) + (\sqrt{2} \cdot 3) + (\sqrt{2} \cdot -\sqrt{2}).$

2. Simplify each term:

   • $6 \cdot 3 = 18$,
   • $6 \cdot -\sqrt{2} = -6\sqrt{2}$,
   • $\sqrt{2} \cdot 3 = 3\sqrt{2}$,
   • $\sqrt{2} \cdot -\sqrt{2} = -2$ (because $\sqrt{2} \cdot \sqrt{2} = 2$).

   So, the expression becomes: $18 - 6\sqrt{2} + 3\sqrt{2} - 2.$

3. Combine like terms:

   • Combine constants: $18 - 2 = 16$,
   • Combine square root terms: $-6\sqrt{2} + 3\sqrt{2} = -3\sqrt{2}$.

   Final expression: $16 - 3\sqrt{2}.$

Final Answer for 22: $16 - 3\sqrt{2}$.

Would you like more details on any part of these solutions or further examples? Here are five related questions to explore:

1. How do you simplify expressions with square roots like $\sqrt{20} + 3\sqrt{5}$?
2. What are the steps for multiplying binomials with radicals?
3. How do you simplify $3\sqrt{6} - \sqrt{24}$?
4. What is the difference between rationalizing and simplifying square roots?
5. Can you explain how to simplify $(2 + \sqrt{3})(2 - \sqrt{3})$?

Tip: Always look for opportunities to simplify square roots first before performing further operations!