Multiplying Polynomials Coloring Activity worksheet for simplifying expressions.

To solve the problem of multiplying polynomials, we will use the distributive property (also known as the FOIL method for binomials). Let's go through each expression step by step.

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Problem 1: \( (a + 4)(a + 7) \)

Using the distributive property:
\[
(a + 4)(a + 7) = a(a + 7) + 4(a + 7)
\]
\[
= a^2 + 7a + 4a + 28
\]
\[
= a^2 + 11a + 28
\]

Answer:
\[
\boxed{a^2 + 11a + 28}
\]

---

Problem 2: \( (a - 7)(a - 3) \)

Using the distributive property:
\[
(a - 7)(a - 3) = a(a - 3) - 7(a - 3)
\]
\[
= a^2 - 3a - 7a + 21
\]
\[
= a^2 - 10a + 21
\]

Answer:
\[
\boxed{a^2 - 10a + 21}
\]

---

Problem 3: \( (a - 7)(a - 5) \)

Using the distributive property:
\[
(a - 7)(a - 5) = a(a - 5) - 7(a - 5)
\]
\[
= a^2 - 5a - 7a + 35
\]
\[
= a^2 - 12a + 35
\]

Answer:
\[
\boxed{a^2 - 12a + 35}
\]

---

Problem 4: \( (3x - 4)(2x - 8) \)

Using the distributive property:
\[
(3x - 4)(2x - 8) = 3x(2x - 8) - 4(2x - 8)
\]
\[
= 6x^2 - 24x - 8x + 32
\]
\[
= 6x^2 - 32x + 32
\]

Answer:
\[
\boxed{6x^2 - 32x + 32}
\]

---

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

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

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

---

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

Using the distributive property:
\[
(x - 1)(x^2 + 6x - 1) = x(x^2 + 6x - 1) - 1(x^2 + 6x - 1)
\]
\[
= x^3 + 6x^2 - x - x^2 - 6x + 1
\]
\[
= x^3 + (6x^2 - x^2) + (-x - 6x) + 1
\]
\[
= x^3 + 5x^2 - 7x + 1
\]

Answer:
\[
\boxed{x^3 + 5x^2 - 7x + 1}
\]

---

Problem 7: \( (x - 8)(x - 2) \)

Using the distributive property:
\[
(x - 8)(x - 2) = x(x - 2) - 8(x - 2)
\]
\[
= x^2 - 2x - 8x + 16
\]
\[
= x^2 - 10x + 16
\]

Answer:
\[
\boxed{x^2 - 10x + 16}
\]

---

Problem 8: \( (x + 6)(x - 3) \)

Using the distributive property:
\[
(x + 6)(x - 3) = x(x - 3) + 6(x - 3)
\]
\[
= x^2 - 3x + 6x - 18
\]
\[
= x^2 + 3x - 18
\]

Answer:
\[
\boxed{x^2 + 3x - 18}
\]

---

Problem 9: \( (4x + 1)(3x - 4) \)

Using the distributive property:
\[
(4x + 1)(3x - 4) = 4x(3x - 4) + 1(3x - 4)
\]
\[
= 12x^2 - 16x + 3x - 4
\]
\[
= 12x^2 - 13x - 4
\]

Answer:
\[
\boxed{12x^2 - 13x - 4}
\]

---

Problem 10: \( (2x - 6)(x + 2) \)

Using the distributive property:
\[
(2x - 6)(x + 2) = 2x(x + 2) - 6(x + 2)
\]
\[
= 2x^2 + 4x - 6x - 12
\]
\[
= 2x^2 - 2x - 12
\]

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

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Final Answers:

1. \( \boxed{a^2 + 11a + 28} \)
2. \( \boxed{a^2 - 10a + 21} \)
3. \( \boxed{a^2 - 12a + 35} \)
4. \( \boxed{6x^2 - 32x + 32} \)
5. \( \boxed{x^2 - x - 2} \)
6. \( \boxed{x^3 + 5x^2 - 7x + 1} \)
7. \( \boxed{x^2 - 10x + 16} \)
8. \( \boxed{x^2 + 3x - 18} \)
9. \( \boxed{12x^2 - 13x - 4} \)
10. \( \boxed{2x^2 - 2x - 12} \)

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Boxed Final Answer:
\[
\boxed{
\begin{aligned}
1. & \ a^2 + 11a + 28 \\
2. & \ a^2 - 10a + 21 \\
3. & \ a^2 - 12a + 35 \\
4. & \ 6x^2 - 32x + 32 \\
5. & \ x^2 - x - 2 \\
6. & \ x^3 + 5x^2 - 7x + 1 \\
7. & \ x^2 - 10x + 16 \\
8. & \ x^2 + 3x - 18 \\
9. & \ 12x^2 - 13x - 4 \\
10. & \ 2x^2 - 2x - 12
\end{aligned}
}
\]