Algebra worksheet for practicing multiplication of binomials, converting products into standard quadratic form ax² + bx + c.

Problem Overview:

The task involves multiplying binomials and writing the resulting expressions in the expanded form \( ax^2 + bx + c \). Each problem requires applying the distributive property (also known as the FOIL method for binomials) to expand the given products.

Solution Approach:

For each pair of binomials, we will use the distributive property to multiply them. The general steps are:
1. Multiply the First terms.
2. Multiply the Outer terms.
3. Multiply the Inner terms.
4. Multiply the Last terms.
5. Combine like terms to get the final expanded expression.

Let's solve each problem step by step.

---

Problem 1: \( (3v - 2)(2v + 5) \)

Using the distributive property:
\[
(3v - 2)(2v + 5) = (3v)(2v) + (3v)(5) + (-2)(2v) + (-2)(5)
\]
\[
= 6v^2 + 15v - 4v - 10
\]
Combine like terms:
\[
= 6v^2 + 11v - 10
\]

Answer: \( 6v^2 + 11v - 10 \)

---

Problem 2: \( (2x - 3)(x + 4) \)

Using the distributive property:
\[
(2x - 3)(x + 4) = (2x)(x) + (2x)(4) + (-3)(x) + (-3)(4)
\]
\[
= 2x^2 + 8x - 3x - 12
\]
Combine like terms:
\[
= 2x^2 + 5x - 12
\]

Answer: \( 2x^2 + 5x - 12 \)

---

Problem 3: \( (2m + 1)(5m + 5) \)

Using the distributive property:
\[
(2m + 1)(5m + 5) = (2m)(5m) + (2m)(5) + (1)(5m) + (1)(5)
\]
\[
= 10m^2 + 10m + 5m + 5
\]
Combine like terms:
\[
= 10m^2 + 15m + 5
\]

Answer: \( 10m^2 + 15m + 5 \)

---

Problem 4: \( (3u + 3)(5u - 1) \)

Using the distributive property:
\[
(3u + 3)(5u - 1) = (3u)(5u) + (3u)(-1) + (3)(5u) + (3)(-1)
\]
\[
= 15u^2 - 3u + 15u - 3
\]
Combine like terms:
\[
= 15u^2 + 12u - 3
\]

Answer: \( 15u^2 + 12u - 3 \)

---

Problem 5: \( (4k + 3)(5k + 1) \)

Using the distributive property:
\[
(4k + 3)(5k + 1) = (4k)(5k) + (4k)(1) + (3)(5k) + (3)(1)
\]
\[
= 20k^2 + 4k + 15k + 3
\]
Combine like terms:
\[
= 20k^2 + 19k + 3
\]

Answer: \( 20k^2 + 19k + 3 \)

---

Problem 6: \( (3x + 3)(5x + 1) \)

Using the distributive property:
\[
(3x + 3)(5x + 1) = (3x)(5x) + (3x)(1) + (3)(5x) + (3)(1)
\]
\[
= 15x^2 + 3x + 15x + 3
\]
Combine like terms:
\[
= 15x^2 + 18x + 3
\]

Answer: \( 15x^2 + 18x + 3 \)

---

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

Using the distributive property:
\[
(5x - 4)(5x + 3) = (5x)(5x) + (5x)(3) + (-4)(5x) + (-4)(3)
\]
\[
= 25x^2 + 15x - 20x - 12
\]
Combine like terms:
\[
= 25x^2 - 5x - 12
\]

Answer: \( 25x^2 - 5x - 12 \)

---

Problem 8: \( (3x + 3)(2x - 1) \)

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

Answer: \( 6x^2 + 3x - 3 \)

---

Problem 9: \( (3n - 3)(3n + 5) \)

Using the distributive property:
\[
(3n - 3)(3n + 5) = (3n)(3n) + (3n)(5) + (-3)(3n) + (-3)(5)
\]
\[
= 9n^2 + 15n - 9n - 15
\]
Combine like terms:
\[
= 9n^2 + 6n - 15
\]

Answer: \( 9n^2 + 6n - 15 \)

---

Problem 10: \( (4x + 2)(x - 1) \)

Using the distributive property:
\[
(4x + 2)(x - 1) = (4x)(x) + (4x)(-1) + (2)(x) + (2)(-1)
\]
\[
= 4x^2 - 4x + 2x - 2
\]
Combine like terms:
\[
= 4x^2 - 2x - 2
\]

Answer: \( 4x^2 - 2x - 2 \)

---

Problem 11: \( (-2n - 6)(5n - 3) \)

Using the distributive property:
\[
(-2n - 6)(5n - 3) = (-2n)(5n) + (-2n)(-3) + (-6)(5n) + (-6)(-3)
\]
\[
= -10n^2 + 6n - 30n + 18
\]
Combine like terms:
\[
= -10n^2 - 24n + 18
\]

Answer: \( -10n^2 - 24n + 18 \)

---

Problem 12: \( (-7h + 7)(-6h - 6) \)

Using the distributive property:
\[
(-7h + 7)(-6h - 6) = (-7h)(-6h) + (-7h)(-6) + (7)(-6h) + (7)(-6)
\]
\[
= 42h^2 + 42h - 42h - 42
\]
Combine like terms:
\[
= 42h^2 - 42
\]

Answer: \( 42h^2 - 42 \)

---

Problem 13: \( (4b - 8)(-8b - 8) \)

Using the distributive property:
\[
(4b - 8)(-8b - 8) = (4b)(-8b) + (4b)(-8) + (-8)(-8b) + (-8)(-8)
\]
\[
= -32b^2 - 32b + 64b + 64
\]
Combine like terms:
\[
= -32b^2 + 32b + 64
\]

Answer: \( -32b^2 + 32b + 64 \)

---

Problem 14: \( (7a - 2)(-2a + 7) \)

Using the distributive property:
\[
(7a - 2)(-2a + 7) = (7a)(-2a) + (7a)(7) + (-2)(-2a) + (-2)(7)
\]
\[
= -14a^2 + 49a + 4a - 14
\]
Combine like terms:
\[
= -14a^2 + 53a - 14
\]

Answer: \( -14a^2 + 53a - 14 \)

---

Problem 15: \( (2b + 5)(-8b + 5) \)

Using the distributive property:
\[
(2b + 5)(-8b + 5) = (2b)(-8b) + (2b)(5) + (5)(-8b) + (5)(5)
\]
\[
= -16b^2 + 10b - 40b + 25
\]
Combine like terms:
\[
= -16b^2 - 30b + 25
\]

Answer: \( -16b^2 - 30b + 25 \)

---

Problem 16: \( (-a - 2)(-7a + 7) \)

Using the distributive property:
\[
(-a - 2)(-7a + 7) = (-a)(-7a) + (-a)(7) + (-2)(-7a) + (-2)(7)
\]
\[
= 7a^2 - 7a + 14a - 14
\]
Combine like terms:
\[
= 7a^2 + 7a - 14
\]

Answer: \( 7a^2 + 7a - 14 \)

---

Problem 17: \( (-8n - 4)(-4n + 3) \)

Using the distributive property:
\[
(-8n - 4)(-4n + 3) = (-8n)(-4n) + (-8n)(3) + (-4)(-4n) + (-4)(3)
\]
\[
= 32n^2 - 24n + 16n - 12
\]
Combine like terms:
\[
= 32n^2 - 8n - 12
\]

Answer: \( 32n^2 - 8n - 12 \)

---

Problem 18: \( (-4n - 6)(4n - 8) \)

Using the distributive property:
\[
(-4n - 6)(4n - 8) = (-4n)(4n) + (-4n)(-8) + (-6)(4n) + (-6)(-8)
\]
\[
= -16n^2 + 32n - 24n + 48
\]
Combine like terms:
\[
= -16n^2 + 8n + 48
\]

Answer: \( -16n^2 + 8n + 48 \)

---

Problem 19: \( (5m - 1)(7m + 8) \)

Using the distributive property:
\[
(5m - 1)(7m + 8) = (5m)(7m) + (5m)(8) + (-1)(7m) + (-1)(8)
\]
\[
= 35m^2 + 40m - 7m - 8
\]
Combine like terms:
\[
= 35m^2 + 33m - 8
\]

Answer: \( 35m^2 + 33m - 8 \)

---

Problem 20: \( (7n - 2)(-n + 8) \)

Using the distributive property:
\[
(7n - 2)(-n + 8) = (7n)(-n) + (7n)(8) + (-2)(-n) + (-2)(8)
\]
\[
= -7n^2 + 56n + 2n - 16
\]
Combine like terms:
\[
= -7n^2 + 58n - 16
\]

Answer: \( -7n^2 + 58n - 16 \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 6v^2 + 11v - 10 \\
2. & \ 2x^2 + 5x - 12 \\
3. & \ 10m^2 + 15m + 5 \\
4. & \ 15u^2 + 12u - 3 \\
5. & \ 20k^2 + 19k + 3 \\
6. & \ 15x^2 + 18x + 3 \\
7. & \ 25x^2 - 5x - 12 \\
8. & \ 6x^2 + 3x - 3 \\
9. & \ 9n^2 + 6n - 15 \\
10. & \ 4x^2 - 2x - 2 \\
11. & \ -10n^2 - 24n + 18 \\
12. & \ 42h^2 - 42 \\
13. & \ -32b^2 + 32b + 64 \\
14. & \ -14a^2 + 53a - 14 \\
15. & \ -16b^2 - 30b + 25 \\
16. & \ 7a^2 + 7a - 14 \\
17. & \ 32n^2 - 8n - 12 \\
18. & \ -16n^2 + 8n + 48 \\
19. & \ 35m^2 + 33m - 8 \\
20. & \ -7n^2 + 58n - 16 \\
\end{aligned}
}
\]