Worksheet titled "Multiplying Radical Expressions Worksheet" from Math Monks, featuring eight problems for students to solve.

To solve the given problems involving multiplying radical expressions, we will use the distributive property (also known as the FOIL method for binomials) and simplify the resulting expressions. Let's solve each problem step by step.

---

Problem 1:

\[
(-4\sqrt{6} + 2)(\sqrt{6} - 5)
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(-4\sqrt{6})(\sqrt{6}) + (-4\sqrt{6})(-5) + (2)(\sqrt{6}) + (2)(-5)
\]

#### Step 2: Simplify each term.
1. \((-4\sqrt{6})(\sqrt{6}) = -4 \cdot (\sqrt{6} \cdot \sqrt{6}) = -4 \cdot 6 = -24\)
2. \((-4\sqrt{6})(-5) = 20\sqrt{6}\)
3. \((2)(\sqrt{6}) = 2\sqrt{6}\)
4. \((2)(-5) = -10\)

#### Step 3: Combine like terms.
\[
-24 + 20\sqrt{6} + 2\sqrt{6} - 10 = -34 + 22\sqrt{6}
\]

#### Final Answer:
\[
\boxed{-34 + 22\sqrt{6}}
\]

---

Problem 2:

\[
-2\sqrt{12}(3 + \sqrt{12})
\]

#### Step 1: Distribute \(-2\sqrt{12}\).
\[
-2\sqrt{12} \cdot 3 + (-2\sqrt{12}) \cdot \sqrt{12}
\]

#### Step 2: Simplify each term.
1. \(-2\sqrt{12} \cdot 3 = -6\sqrt{12}\)
2. \((-2\sqrt{12}) \cdot \sqrt{12} = -2 \cdot (\sqrt{12} \cdot \sqrt{12}) = -2 \cdot 12 = -24\)

#### Step 3: Combine the terms.
\[
-6\sqrt{12} - 24
\]

#### Step 4: Simplify \(\sqrt{12}\).
\[
\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}
\]
So, \(-6\sqrt{12} = -6 \cdot 2\sqrt{3} = -12\sqrt{3}\).

#### Final Answer:
\[
\boxed{-12\sqrt{3} - 24}
\]

---

Problem 3:

\[
(5 - 4\sqrt{5})(-2 + \sqrt{5})
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(5)(-2) + (5)(\sqrt{5}) + (-4\sqrt{5})(-2) + (-4\sqrt{5})(\sqrt{5})
\]

#### Step 2: Simplify each term.
1. \((5)(-2) = -10\)
2. \((5)(\sqrt{5}) = 5\sqrt{5}\)
3. \((-4\sqrt{5})(-2) = 8\sqrt{5}\)
4. \((-4\sqrt{5})(\sqrt{5}) = -4 \cdot (\sqrt{5} \cdot \sqrt{5}) = -4 \cdot 5 = -20\)

#### Step 3: Combine like terms.
\[
-10 + 5\sqrt{5} + 8\sqrt{5} - 20 = -30 + 13\sqrt{5}
\]

#### Final Answer:
\[
\boxed{-30 + 13\sqrt{5}}
\]

---

Problem 4:

\[
(5 + 4\sqrt{3})(3 + \sqrt{3})
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(5)(3) + (5)(\sqrt{3}) + (4\sqrt{3})(3) + (4\sqrt{3})(\sqrt{3})
\]

#### Step 2: Simplify each term.
1. \((5)(3) = 15\)
2. \((5)(\sqrt{3}) = 5\sqrt{3}\)
3. \((4\sqrt{3})(3) = 12\sqrt{3}\)
4. \((4\sqrt{3})(\sqrt{3}) = 4 \cdot (\sqrt{3} \cdot \sqrt{3}) = 4 \cdot 3 = 12\)

#### Step 3: Combine like terms.
\[
15 + 5\sqrt{3} + 12\sqrt{3} + 12 = 27 + 17\sqrt{3}
\]

#### Final Answer:
\[
\boxed{27 + 17\sqrt{3}}
\]

---

Problem 5:

\[
(1 - 4\sqrt{2})(4 + \sqrt{2})
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(1)(4) + (1)(\sqrt{2}) + (-4\sqrt{2})(4) + (-4\sqrt{2})(\sqrt{2})
\]

#### Step 2: Simplify each term.
1. \((1)(4) = 4\)
2. \((1)(\sqrt{2}) = \sqrt{2}\)
3. \((-4\sqrt{2})(4) = -16\sqrt{2}\)
4. \((-4\sqrt{2})(\sqrt{2}) = -4 \cdot (\sqrt{2} \cdot \sqrt{2}) = -4 \cdot 2 = -8\)

#### Step 3: Combine like terms.
\[
4 + \sqrt{2} - 16\sqrt{2} - 8 = -4 - 15\sqrt{2}
\]

#### Final Answer:
\[
\boxed{-4 - 15\sqrt{2}}
\]

---

Problem 6:

\[
(2\sqrt{3} - 2)(\sqrt{3} - 1)
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(2\sqrt{3})(\sqrt{3}) + (2\sqrt{3})(-1) + (-2)(\sqrt{3}) + (-2)(-1)
\]

#### Step 2: Simplify each term.
1. \((2\sqrt{3})(\sqrt{3}) = 2 \cdot (\sqrt{3} \cdot \sqrt{3}) = 2 \cdot 3 = 6\)
2. \((2\sqrt{3})(-1) = -2\sqrt{3}\)
3. \((-2)(\sqrt{3}) = -2\sqrt{3}\)
4. \((-2)(-1) = 2\)

#### Step 3: Combine like terms.
\[
6 - 2\sqrt{3} - 2\sqrt{3} + 2 = 8 - 4\sqrt{3}
\]

#### Final Answer:
\[
\boxed{8 - 4\sqrt{3}}
\]

---

Problem 7:

\[
(-4\sqrt{2x} + 2)(-2\sqrt{2} - 4)
\]

#### Step 1: Apply the distributive property (FOIL method).
\[
(-4\sqrt{2x})(-2\sqrt{2}) + (-4\sqrt{2x})(-4) + (2)(-2\sqrt{2}) + (2)(-4)
\]

#### Step 2: Simplify each term.
1. \((-4\sqrt{2x})(-2\sqrt{2}) = 8\sqrt{2x \cdot 2} = 8\sqrt{4x} = 8 \cdot 2\sqrt{x} = 16\sqrt{x}\)
2. \((-4\sqrt{2x})(-4) = 16\sqrt{2x}\)
3. \((2)(-2\sqrt{2}) = -4\sqrt{2}\)
4. \((2)(-4) = -8\)

#### Step 3: Combine like terms.
\[
16\sqrt{x} + 16\sqrt{2x} - 4\sqrt{2} - 8
\]

#### Final Answer:
\[
\boxed{16\sqrt{x} + 16\sqrt{2x} - 4\sqrt{2} - 8}
\]

---

Problem 8:

\[
(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})
\]

#### Step 1: Recognize the difference of squares pattern.
\[
(a - b)(a + b) = a^2 - b^2
\]
Here, \(a = \sqrt{5}\) and \(b = \sqrt{3}\).

#### Step 2: Apply the formula.
\[
(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2
\]

#### Final Answer:
\[
\boxed{2}
\]

---

Final Answers:

1. \(\boxed{-34 + 22\sqrt{6}}\)
2. \(\boxed{-12\sqrt{3} - 24}\)
3. \(\boxed{-30 + 13\sqrt{5}}\)
4. \(\boxed{27 + 17\sqrt{3}}\)
5. \(\boxed{-4 - 15\sqrt{2}}\)
6. \(\boxed{8 - 4\sqrt{3}}\)
7. \(\boxed{16\sqrt{x} + 16\sqrt{2x} - 4\sqrt{2} - 8}\)
8. \(\boxed{2}\)