To simplify the expression $(3\sqrt{2} + \sqrt{5})(2\sqrt{3} - \sqrt{4})$, let’s proceed step by step.

---

Given:

$(3\sqrt{2} + \sqrt{5})(2\sqrt{3} - \sqrt{4})$

1. Expand the product using the distributive property: $= (3\sqrt{2})(2\sqrt{3}) + (3\sqrt{2})(-\sqrt{4}) + (\sqrt{5})(2\sqrt{3}) + (\sqrt{5})(-\sqrt{4})$

2. Simplify each term:

   • $(3\sqrt{2})(2\sqrt{3}) = 6\sqrt{6}$
   • $(3\sqrt{2})(-\sqrt{4}) = -3\sqrt{2} \cdot 2 = -6\sqrt{2}$
   • $(\sqrt{5})(2\sqrt{3}) = 2\sqrt{15}$
   • $(\sqrt{5})(-\sqrt{4}) = -\sqrt{5} \cdot 2 = -2\sqrt{5}$

3. Combine all terms: $= 6\sqrt{6} - 6\sqrt{2} + 2\sqrt{15} - 2\sqrt{5}$

4. Final expression: $6\sqrt{6} + 2\sqrt{15} - 6\sqrt{2} - 2\sqrt{5}$

---

This is the simplified form of the given expression.

Would you like more details or a deeper explanation?

---

Related Questions:

1. What is the importance of simplifying radical expressions in mathematics?
2. How does the distributive property apply to products involving radicals?
3. Can this expression be factored further or simplified more?
4. What are some common mistakes made when multiplying radicals?
5. How do the rules of exponents relate to operations with radicals?

Tip:

When working with radicals, always ensure that any square roots are simplified fully before combining like terms.