Multiplying Binomials worksheet with algebraic expressions to solve.

To solve the problem of multiplying binomials, we will use the distributive property (also known as the FOIL method for binomials). The FOIL method stands for:

- First: Multiply the first terms in each binomial.
- Outer: Multiply the outer terms in the product.
- Inner: Multiply the inner terms in the product.
- Last: Multiply the last terms in each binomial.

We will apply this method to each of the given problems step by step.

---

Problem 1: \( (3n + 2)(n + 3) \)

Using the distributive property:
\[
(3n + 2)(n + 3) = 3n \cdot n + 3n \cdot 3 + 2 \cdot n + 2 \cdot 3
\]
\[
= 3n^2 + 9n + 2n + 6
\]
Combine like terms:
\[
= 3n^2 + 11n + 6
\]

Answer:
\[
\boxed{3n^2 + 11n + 6}
\]

---

Problem 2: \( (3p - 3)(p - 1) \)

Using the distributive property:
\[
(3p - 3)(p - 1) = 3p \cdot p + 3p \cdot (-1) + (-3) \cdot p + (-3) \cdot (-1)
\]
\[
= 3p^2 - 3p - 3p + 3
\]
Combine like terms:
\[
= 3p^2 - 6p + 3
\]

Answer:
\[
\boxed{3p^2 - 6p + 3}
\]

---

Problem 3: \( (2x + 1)(x - 1) \)

Using the distributive property:
\[
(2x + 1)(x - 1) = 2x \cdot x + 2x \cdot (-1) + 1 \cdot x + 1 \cdot (-1)
\]
\[
= 2x^2 - 2x + x - 1
\]
Combine like terms:
\[
= 2x^2 - x - 1
\]

Answer:
\[
\boxed{2x^2 - x - 1}
\]

---

Problem 4: \( (5x - 2)(5x - 8) \)

Using the distributive property:
\[
(5x - 2)(5x - 8) = 5x \cdot 5x + 5x \cdot (-8) + (-2) \cdot 5x + (-2) \cdot (-8)
\]
\[
= 25x^2 - 40x - 10x + 16
\]
Combine like terms:
\[
= 25x^2 - 50x + 16
\]

Answer:
\[
\boxed{25x^2 - 50x + 16}
\]

---

Problem 5: \( (5v + 4)(3v - 6) \)

Using the distributive property:
\[
(5v + 4)(3v - 6) = 5v \cdot 3v + 5v \cdot (-6) + 4 \cdot 3v + 4 \cdot (-6)
\]
\[
= 15v^2 - 30v + 12v - 24
\]
Combine like terms:
\[
= 15v^2 - 18v - 24
\]

Answer:
\[
\boxed{15v^2 - 18v - 24}
\]

---

Problem 6: \( (x + 1)(x + 2) \)

Using the distributive property:
\[
(x + 1)(x + 2) = x \cdot x + x \cdot 2 + 1 \cdot x + 1 \cdot 2
\]
\[
= x^2 + 2x + x + 2
\]
Combine like terms:
\[
= x^2 + 3x + 2
\]

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

---

Problem 7: \( (x - 5)(x - 4) \)

Using the distributive property:
\[
(x - 5)(x - 4) = x \cdot x + x \cdot (-4) + (-5) \cdot x + (-5) \cdot (-4)
\]
\[
= x^2 - 4x - 5x + 20
\]
Combine like terms:
\[
= x^2 - 9x + 20
\]

Answer:
\[
\boxed{x^2 - 9x + 20}
\]

---

Problem 8: \( (x + 5)(3x - 1) \)

Using the distributive property:
\[
(x + 5)(3x - 1) = x \cdot 3x + x \cdot (-1) + 5 \cdot 3x + 5 \cdot (-1)
\]
\[
= 3x^2 - x + 15x - 5
\]
Combine like terms:
\[
= 3x^2 + 14x - 5
\]

Answer:
\[
\boxed{3x^2 + 14x - 5}
\]

---

Problem 9: \( (3x - y)(x + 2y) \)

Using the distributive property:
\[
(3x - y)(x + 2y) = 3x \cdot x + 3x \cdot 2y + (-y) \cdot x + (-y) \cdot 2y
\]
\[
= 3x^2 + 6xy - xy - 2y^2
\]
Combine like terms:
\[
= 3x^2 + 5xy - 2y^2
\]

Answer:
\[
\boxed{3x^2 + 5xy - 2y^2}
\]

---

Problem 10: \( (4x - 5)(x - 3) \)

Using the distributive property:
\[
(4x - 5)(x - 3) = 4x \cdot x + 4x \cdot (-3) + (-5) \cdot x + (-5) \cdot (-3)
\]
\[
= 4x^2 - 12x - 5x + 15
\]
Combine like terms:
\[
= 4x^2 - 17x + 15
\]

Answer:
\[
\boxed{4x^2 - 17x + 15}
\]

---

Problem 11: \( (x - 1)(2x + 5) \)

Using the distributive property:
\[
(x - 1)(2x + 5) = x \cdot 2x + x \cdot 5 + (-1) \cdot 2x + (-1) \cdot 5
\]
\[
= 2x^2 + 5x - 2x - 5
\]
Combine like terms:
\[
= 2x^2 + 3x - 5
\]

Answer:
\[
\boxed{2x^2 + 3x - 5}
\]

---

Problem 12: \( (3x - 3)(3x + 2) \)

Using the distributive property:
\[
(3x - 3)(3x + 2) = 3x \cdot 3x + 3x \cdot 2 + (-3) \cdot 3x + (-3) \cdot 2
\]
\[
= 9x^2 + 6x - 9x - 6
\]
Combine like terms:
\[
= 9x^2 - 3x - 6
\]

Answer:
\[
\boxed{9x^2 - 3x - 6}
\]

---

Problem 13: \( (6x + 2y)(6x - 2y) \)

This is a difference of squares:
\[
(a + b)(a - b) = a^2 - b^2
\]
Here, \( a = 6x \) and \( b = 2y \):
\[
(6x + 2y)(6x - 2y) = (6x)^2 - (2y)^2
\]
\[
= 36x^2 - 4y^2
\]

Answer:
\[
\boxed{36x^2 - 4y^2}
\]

---

Problem 14: \( (5x - 7)(3x - 4) \)

Using the distributive property:
\[
(5x - 7)(3x - 4) = 5x \cdot 3x + 5x \cdot (-4) + (-7) \cdot 3x + (-7) \cdot (-4)
\]
\[
= 15x^2 - 20x - 21x + 28
\]
Combine like terms:
\[
= 15x^2 - 41x + 28
\]

Answer:
\[
\boxed{15x^2 - 41x + 28}
\]

---

Problem 15: \( (-2x + 5)(x - 1) \)

Using the distributive property:
\[
(-2x + 5)(x - 1) = -2x \cdot x + (-2x) \cdot (-1) + 5 \cdot x + 5 \cdot (-1)
\]
\[
= -2x^2 + 2x + 5x - 5
\]
Combine like terms:
\[
= -2x^2 + 7x - 5
\]

Answer:
\[
\boxed{-2x^2 + 7x - 5}
\]

---

Final Answers:

1. \( \boxed{3n^2 + 11n + 6} \)
2. \( \boxed{3p^2 - 6p + 3} \)
3. \( \boxed{2x^2 - x - 1} \)
4. \( \boxed{25x^2 - 50x + 16} \)
5. \( \boxed{15v^2 - 18v - 24} \)
6. \( \boxed{x^2 + 3x + 2} \)
7. \( \boxed{x^2 - 9x + 20} \)
8. \( \boxed{3x^2 + 14x - 5} \)
9. \( \boxed{3x^2 + 5xy - 2y^2} \)
10. \( \boxed{4x^2 - 17x + 15} \)
11. \( \boxed{2x^2 + 3x - 5} \)
12. \( \boxed{9x^2 - 3x - 6} \)
13. \( \boxed{36x^2 - 4y^2} \)
14. \( \boxed{15x^2 - 41x + 28} \)
15. \( \boxed{-2x^2 + 7x - 5} \)

Boxed Final Answer:
\[
\boxed{
\begin{aligned}
&1. 3n^2 + 11n + 6 \\
&2. 3p^2 - 6p + 3 \\
&3. 2x^2 - x - 1 \\
&4. 25x^2 - 50x + 16 \\
&5. 15v^2 - 18v - 24 \\
&6. x^2 + 3x + 2 \\
&7. x^2 - 9x + 20 \\
&8. 3x^2 + 14x - 5 \\
&9. 3x^2 + 5xy - 2y^2 \\
&10. 4x^2 - 17x + 15 \\
&11. 2x^2 + 3x - 5 \\
&12. 9x^2 - 3x - 6 \\
&13. 36x^2 - 4y^2 \\
&14. 15x^2 - 41x + 28 \\
&15. -2x^2 + 7x - 5
\end{aligned}
}
\]