Multiplying Binomials worksheet with problems and answers.

Problem: Multiplying Binomials

The task involves multiplying pairs of binomials and simplifying the resulting expressions. Below, I will solve each problem step by step and explain the process.

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#### 1. \( (2x + y)(x - 2y) \)

Step 1: Use the distributive property (also known as the FOIL method for binomials):
\[
(2x + y)(x - 2y) = 2x \cdot x + 2x \cdot (-2y) + y \cdot x + y \cdot (-2y)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - 4xy + xy - 2y^2
\]

Step 3: Combine like terms:
\[
= 2x^2 - 3xy - 2y^2
\]

Final Answer:
\[
\boxed{2x^2 - 3xy - 2y^2}
\]

---

#### 2. \( (x - 3y)(2x - y) \)

Step 1: Use the distributive property:
\[
(x - 3y)(2x - y) = x \cdot 2x + x \cdot (-y) + (-3y) \cdot 2x + (-3y) \cdot (-y)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - xy - 6xy + 3y^2
\]

Step 3: Combine like terms:
\[
= 2x^2 - 7xy + 3y^2
\]

Final Answer:
\[
\boxed{2x^2 - 7xy + 3y^2}
\]

---

#### 3. \( (x - 4)(2x - 1) \)

Step 1: Use the distributive property:
\[
(x - 4)(2x - 1) = x \cdot 2x + x \cdot (-1) + (-4) \cdot 2x + (-4) \cdot (-1)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - x - 8x + 4
\]

Step 3: Combine like terms:
\[
= 2x^2 - 9x + 4
\]

Final Answer:
\[
\boxed{2x^2 - 9x + 4}
\]

---

#### 4. \( (x + 2)(2x + 3) \)

Step 1: Use the distributive property:
\[
(x + 2)(2x + 3) = x \cdot 2x + x \cdot 3 + 2 \cdot 2x + 2 \cdot 3
\]

Step 2: Perform the multiplications:
\[
= 2x^2 + 3x + 4x + 6
\]

Step 3: Combine like terms:
\[
= 2x^2 + 7x + 6
\]

Final Answer:
\[
\boxed{2x^2 + 7x + 6}
\]

---

#### 5. \( (2x + 4)(x - 3) \)

Step 1: Use the distributive property:
\[
(2x + 4)(x - 3) = 2x \cdot x + 2x \cdot (-3) + 4 \cdot x + 4 \cdot (-3)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - 6x + 4x - 12
\]

Step 3: Combine like terms:
\[
= 2x^2 - 2x - 12
\]

Final Answer:
\[
\boxed{2x^2 - 2x - 12}
\]

---

#### 6. \( (2x + 1)(x + 2) \)

Step 1: Use the distributive property:
\[
(2x + 1)(x + 2) = 2x \cdot x + 2x \cdot 2 + 1 \cdot x + 1 \cdot 2
\]

Step 2: Perform the multiplications:
\[
= 2x^2 + 4x + x + 2
\]

Step 3: Combine like terms:
\[
= 2x^2 + 5x + 2
\]

Final Answer:
\[
\boxed{2x^2 + 5x + 2}
\]

---

#### 7. \( (3x - 3y)(2x - y) \)

Step 1: Use the distributive property:
\[
(3x - 3y)(2x - y) = 3x \cdot 2x + 3x \cdot (-y) + (-3y) \cdot 2x + (-3y) \cdot (-y)
\]

Step 2: Perform the multiplications:
\[
= 6x^2 - 3xy - 6xy + 3y^2
\]

Step 3: Combine like terms:
\[
= 6x^2 - 9xy + 3y^2
\]

Final Answer:
\[
\boxed{6x^2 - 9xy + 3y^2}
\]

---

#### 8. \( (x - 3)(2x - 1) \)

Step 1: Use the distributive property:
\[
(x - 3)(2x - 1) = x \cdot 2x + x \cdot (-1) + (-3) \cdot 2x + (-3) \cdot (-1)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - x - 6x + 3
\]

Step 3: Combine like terms:
\[
= 2x^2 - 7x + 3
\]

Final Answer:
\[
\boxed{2x^2 - 7x + 3}
\]

---

#### 9. \( (x + 2y)(2x - y) \)

Step 1: Use the distributive property:
\[
(x + 2y)(2x - y) = x \cdot 2x + x \cdot (-y) + 2y \cdot 2x + 2y \cdot (-y)
\]

Step 2: Perform the multiplications:
\[
= 2x^2 - xy + 4xy - 2y^2
\]

Step 3: Combine like terms:
\[
= 2x^2 + 3xy - 2y^2
\]

Final Answer:
\[
\boxed{2x^2 + 3xy - 2y^2}
\]

---

#### 10. \( (2x + 1)(x + 2) \)

Step 1: Use the distributive property:
\[
(2x + 1)(x + 2) = 2x \cdot x + 2x \cdot 2 + 1 \cdot x + 1 \cdot 2
\]

Step 2: Perform the multiplications:
\[
= 2x^2 + 4x + x + 2
\]

Step 3: Combine like terms:
\[
= 2x^2 + 5x + 2
\]

Final Answer:
\[
\boxed{2x^2 + 5x + 2}
\]

---

Summary of Answers:

1. \( 2x^2 - 3xy - 2y^2 \)
2. \( 2x^2 - 7xy + 3y^2 \)
3. \( 2x^2 - 9x + 4 \)
4. \( 2x^2 + 7x + 6 \)
5. \( 2x^2 - 2x - 12 \)
6. \( 2x^2 + 5x + 2 \)
7. \( 6x^2 - 9xy + 3y^2 \)
8. \( 2x^2 - 7x + 3 \)
9. \( 2x^2 + 3xy - 2y^2 \)
10. \( 2x^2 + 5x + 2 \)

\boxed{\text{All answers are correct.}}