Worksheet for practicing multiplication of binomials by trinomials.

Problem: Simplify each expression by multiplying a binomial by a trinomial.

We will solve each problem step by step using the distributive property (also known as the FOIL method extended to trinomials). The general approach is to distribute each term in the binomial to every term in the trinomial.

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#### 1. \( (6c^4 - c^3)(9c^3 - 7c^2 - 3c) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(6c^4)(9c^3) + (6c^4)(-7c^2) + (6c^4)(-3c) + (-c^3)(9c^3) + (-c^3)(-7c^2) + (-c^3)(-3c)
\]

Calculate each product:

\[
54c^7 - 42c^6 - 18c^5 - 9c^6 + 7c^5 + 3c^4
\]

Combine like terms:

\[
54c^7 + (-42c^6 - 9c^6) + (-18c^5 + 7c^5) + 3c^4
\]

\[
54c^7 - 51c^6 - 11c^5 + 3c^4
\]

Final Answer:

\[
\boxed{54c^7 - 51c^6 - 11c^5 + 3c^4}
\]

---

#### 2. \( (a^4 + 5a^3)(5a^4 - 4a^3 - 5a^2) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(a^4)(5a^4) + (a^4)(-4a^3) + (a^4)(-5a^2) + (5a^3)(5a^4) + (5a^3)(-4a^3) + (5a^3)(-5a^2)
\]

Calculate each product:

\[
5a^8 - 4a^7 - 5a^6 + 25a^7 - 20a^6 - 25a^5
\]

Combine like terms:

\[
5a^8 + (-4a^7 + 25a^7) + (-5a^6 - 20a^6) - 25a^5
\]

\[
5a^8 + 21a^7 - 25a^6 - 25a^5
\]

Final Answer:

\[
\boxed{5a^8 + 21a^7 - 25a^6 - 25a^5}
\]

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#### 3. \( (7k^2 - 9k)(9k^4 - 2k^3 + 3k^2) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(7k^2)(9k^4) + (7k^2)(-2k^3) + (7k^2)(3k^2) + (-9k)(9k^4) + (-9k)(-2k^3) + (-9k)(3k^2)
\]

Calculate each product:

\[
63k^6 - 14k^5 + 21k^4 - 81k^5 + 18k^4 - 27k^3
\]

Combine like terms:

\[
63k^6 + (-14k^5 - 81k^5) + (21k^4 + 18k^4) - 27k^3
\]

\[
63k^6 - 95k^5 + 39k^4 - 27k^3
\]

Final Answer:

\[
\boxed{63k^6 - 95k^5 + 39k^4 - 27k^3}
\]

---

#### 4. \( (2q^2 - q)(2q^2 + 9q - 2) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(2q^2)(2q^2) + (2q^2)(9q) + (2q^2)(-2) + (-q)(2q^2) + (-q)(9q) + (-q)(-2)
\]

Calculate each product:

\[
4q^4 + 18q^3 - 4q^2 - 2q^3 - 9q^2 + 2q
\]

Combine like terms:

\[
4q^4 + (18q^3 - 2q^3) + (-4q^2 - 9q^2) + 2q
\]

\[
4q^4 + 16q^3 - 13q^2 + 2q
\]

Final Answer:

\[
\boxed{4q^4 + 16q^3 - 13q^2 + 2q}
\]

---

#### 5. \( (-8g^4 - 6g^3)(-9g^3 + 2g^2 + 7g) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(-8g^4)(-9g^3) + (-8g^4)(2g^2) + (-8g^4)(7g) + (-6g^3)(-9g^3) + (-6g^3)(2g^2) + (-6g^3)(7g)
\]

Calculate each product:

\[
72g^7 - 16g^6 - 56g^5 + 54g^6 - 12g^5 - 42g^4
\]

Combine like terms:

\[
72g^7 + (-16g^6 + 54g^6) + (-56g^5 - 12g^5) - 42g^4
\]

\[
72g^7 + 38g^6 - 68g^5 - 42g^4
\]

Final Answer:

\[
\boxed{72g^7 + 38g^6 - 68g^5 - 42g^4}
\]

---

#### 6. \( (-5v^2 - b)(9b^3 - b^2 - 2b) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(-5v^2)(9b^3) + (-5v^2)(-b^2) + (-5v^2)(-2b) + (-b)(9b^3) + (-b)(-b^2) + (-b)(-2b)
\]

Calculate each product:

\[
-45v^2b^3 + 5v^2b^2 + 10v^2b - 9b^4 + b^3 + 2b^2
\]

Since there are no like terms, the expression is already simplified.

Final Answer:

\[
\boxed{-45v^2b^3 + 5v^2b^2 + 10v^2b - 9b^4 + b^3 + 2b^2}
\]

---

#### 7. \( (-k + 8)(-2k^2 - 8k - 1) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(-k)(-2k^2) + (-k)(-8k) + (-k)(-1) + (8)(-2k^2) + (8)(-8k) + (8)(-1)
\]

Calculate each product:

\[
2k^3 + 8k^2 + k - 16k^2 - 64k - 8
\]

Combine like terms:

\[
2k^3 + (8k^2 - 16k^2) + (k - 64k) - 8
\]

\[
2k^3 - 8k^2 - 63k - 8
\]

Final Answer:

\[
\boxed{2k^3 - 8k^2 - 63k - 8}
\]

---

#### 8. \( (-7s^4 - 6s^3)(8s^2 - 7s + 4) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(-7s^4)(8s^2) + (-7s^4)(-7s) + (-7s^4)(4) + (-6s^3)(8s^2) + (-6s^3)(-7s) + (-6s^3)(4)
\]

Calculate each product:

\[
-56s^6 + 49s^5 - 28s^4 - 48s^5 + 42s^4 - 24s^3
\]

Combine like terms:

\[
-56s^6 + (49s^5 - 48s^5) + (-28s^4 + 42s^4) - 24s^3
\]

\[
-56s^6 + s^5 + 14s^4 - 24s^3
\]

Final Answer:

\[
\boxed{-56s^6 + s^5 + 14s^4 - 24s^3}
\]

---

#### 9. \( (z^4 + 3z^3)(6z^2 - 4z^2 + 4z) \)

Simplify the trinomial first:

\[
6z^2 - 4z^2 + 4z = 2z^2 + 4z
\]

Now distribute each term in the binomial to each term in the simplified trinomial:

\[
(z^4)(2z^2) + (z^4)(4z) + (3z^3)(2z^2) + (3z^3)(4z)
\]

Calculate each product:

\[
2z^6 + 4z^5 + 6z^5 + 12z^4
\]

Combine like terms:

\[
2z^6 + (4z^5 + 6z^5) + 12z^4
\]

\[
2z^6 + 10z^5 + 12z^4
\]

Final Answer:

\[
\boxed{2z^6 + 10z^5 + 12z^4}
\]

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#### 10. \( (2g^3 + g^2)(4g^3 - 7g^2 - 3g) \)

Distribute each term in the binomial to each term in the trinomial:

\[
(2g^3)(4g^3) + (2g^3)(-7g^2) + (2g^3)(-3g) + (g^2)(4g^3) + (g^2)(-7g^2) + (g^2)(-3g)
\]

Calculate each product:

\[
8g^6 - 14g^5 - 6g^4 + 4g^5 - 7g^4 - 3g^3
\]

Combine like terms:

\[
8g^6 + (-14g^5 + 4g^5) + (-6g^4 - 7g^4) - 3g^3
\]

\[
8g^6 - 10g^5 - 13g^4 - 3g^3
\]

Final Answer:

\[
\boxed{8g^6 - 10g^5 - 13g^4 - 3g^3}
\]

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Final Answers for All Problems:

1. \(\boxed{54c^7 - 51c^6 - 11c^5 + 3c^4}\)
2. \(\boxed{5a^8 + 21a^7 - 25a^6 - 25a^5}\)
3. \(\boxed{63k^6 - 95k^5 + 39k^4 - 27k^3}\)
4. \(\boxed{4q^4 + 16q^3 - 13q^2 + 2q}\)
5. \(\boxed{72g^7 + 38g^6 - 68g^5 - 42g^4}\)
6. \(\boxed{-45v^2b^3 + 5v^2b^2 + 10v^2b - 9b^4 + b^3 + 2b^2}\)
7. \(\boxed{2k^3 - 8k^2 - 63k - 8}\)
8. \(\boxed{-56s^6 + s^5 + 14s^4 - 24s^3}\)
9. \(\boxed{2z^6 + 10z^5 + 12z^4}\)
10. \(\boxed{8g^6 - 10g^5 - 13g^4 - 3g^3}\)