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Let's work through each of the five expressions one by one to simplify them.

### 1. \( 3(5\sqrt{2}) \)
This is a straightforward multiplication:
\[
3 \times 5\sqrt{2} = 15\sqrt{2}
\]
**Simplified expression**: \( 15\sqrt{2} \)

---

### 2. \( (2\sqrt{3})(\sqrt{15}) \)
We can combine the square roots:
\[
2\sqrt{3} \times \sqrt{15} = 2\sqrt{3 \times 15} = 2\sqrt{45}
\]
Now, simplify \( \sqrt{45} \):
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
So, the entire expression becomes:
\[
2 \times 3\sqrt{5} = 6\sqrt{5}
\]
**Simplified expression**: \( 6\sqrt{5} \)

---

### 3. \( 3(2\sqrt{3} - 4\sqrt{5}) \)
Distribute the 3:
\[
3 \times 2\sqrt{3} = 6\sqrt{3}, \quad 3 \times -4\sqrt{5} = -12\sqrt{5}
\]
So, the simplified expression is:
\[
6\sqrt{3} - 12\sqrt{5}
\]
**Simplified expression**: \( 6\sqrt{3} - 12\sqrt{5} \)

---

### 4. \( \sqrt{3}(5\sqrt{6} + 4\sqrt{3}) \)
Distribute the \( \sqrt{3} \):
\[
\sqrt{3} \times 5\sqrt{6} = 5\sqrt{3 \times 6} = 5\sqrt{18} = 5 \times 3\sqrt{2} = 15\sqrt{2}
\]
\[
\sqrt{3} \times 4\sqrt{3} = 4\sqrt{3 \times 3} = 4\sqrt{9} = 4 \times 3 = 12
\]
So, the expression becomes:
\[
15\sqrt{2} + 12
\]
**Simplified expression**: \( 15\sqrt{2} + 12 \)

---

### 5. \( (1 + \sqrt{2})(3 + \sqrt{2}) \)
Apply the distributive property (FOIL method):
\[
1 \times 3 = 3, \quad 1 \times \sqrt{2} = \sqrt{2}
\]
\[
\sqrt{2} \times 3 = 3\sqrt{2}, \quad \sqrt{2} \times \sqrt{2} = 2
\]
Now, add everything:
\[
3 + \sqrt{2} + 3\sqrt{2} + 2 = 5 + 4\sqrt{2}
\]
**Simplified expression**: \( 5 + 4\sqrt{2} \)

---

If you need more details or further clarifications on any step, feel free to ask!

### Relative Questions:
1. How do you simplify the multiplication of two square roots in general?
2. What is the distributive property, and how is it applied to square roots?
3. Why does \( \sqrt{2} \times \sqrt{2} = 2 \) hold true?
4. How do you simplify complex expressions with multiple square roots?
5. What techniques can be used to simplify \( \sqrt{45} \) into \( 3\sqrt{5} \)?

### Tip:
When simplifying square roots, always check if the radicand (number inside the square root) can be factored into a perfect square, which allows for further simplification!

Simplify the following expressions:
1. 3(5√2)
2. (2√3)(√15)
3. 3(2√3 - 4√5)
4. √3(5√6 + 4√3)
5. (1 + √2)(3 + √2)

Learn how to simplify expressions involving multiplication and distribution of square roots. This guide walks through each step, demonstrating techniques such as the distributive property and combining like terms. Suitable for students in Grades 8-10.

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