Printable math worksheet for practicing multiplication of binomials with multi-variable expressions.

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

---

Problem 1:

\[
\left( \frac{1}{3}a^2b + 9 \right) \left( \frac{1}{3}a^2b + 9 \right)
\]

This is a binomial squared. We can use the formula:
\[
(x + y)^2 = x^2 + 2xy + y^2
\]
where \( x = \frac{1}{3}a^2b \) and \( y = 9 \).

1. Compute \( x^2 \):
\[
\left( \frac{1}{3}a^2b \right)^2 = \frac{1}{9}a^4b^2
\]

2. Compute \( 2xy \):
\[
2 \left( \frac{1}{3}a^2b \right)(9) = 2 \cdot \frac{1}{3} \cdot 9 \cdot a^2b = 6a^2b
\]

3. Compute \( y^2 \):
\[
9^2 = 81
\]

Combine all terms:
\[
\frac{1}{9}a^4b^2 + 6a^2b + 81
\]

Answer:
\[
\boxed{\frac{1}{9}a^4b^2 + 6a^2b + 81}
\]

---

Problem 2:

\[
(11vw - 2)(2 + 11vw)
\]

Use the distributive property (FOIL method):
\[
(11vw - 2)(2 + 11vw) = (11vw)(2) + (11vw)(11vw) + (-2)(2) + (-2)(11vw)
\]

1. Compute each term:
\[
(11vw)(2) = 22vw
\]
\[
(11vw)(11vw) = 121v^2w^2
\]
\[
(-2)(2) = -4
\]
\[
(-2)(11vw) = -22vw
\]

Combine all terms:
\[
121v^2w^2 + 22vw - 22vw - 4 = 121v^2w^2 - 4
\]

Answer:
\[
\boxed{121v^2w^2 - 4}
\]

---

Problem 3:

\[
(20rs - 8tu)(-1 - 4s)
\]

Use the distributive property:
\[
(20rs - 8tu)(-1 - 4s) = (20rs)(-1) + (20rs)(-4s) + (-8tu)(-1) + (-8tu)(-4s)
\]

1. Compute each term:
\[
(20rs)(-1) = -20rs
\]
\[
(20rs)(-4s) = -80rs^2
\]
\[
(-8tu)(-1) = 8tu
\]
\[
(-8tu)(-4s) = 32tus
\]

Combine all terms:
\[
-20rs - 80rs^2 + 8tu + 32tus
\]

Answer:
\[
\boxed{-80rs^2 - 20rs + 32tus + 8tu}
\]

---

Problem 4:

\[
(-12x^3 - 6x^2yz)(-6yz - 3x)
\]

Use the distributive property:
\[
(-12x^3 - 6x^2yz)(-6yz - 3x) = (-12x^3)(-6yz) + (-12x^3)(-3x) + (-6x^2yz)(-6yz) + (-6x^2yz)(-3x)
\]

1. Compute each term:
\[
(-12x^3)(-6yz) = 72x^3yz
\]
\[
(-12x^3)(-3x) = 36x^4
\]
\[
(-6x^2yz)(-6yz) = 36x^2y^2z^2
\]
\[
(-6x^2yz)(-3x) = 18x^3yz
\]

Combine all terms:
\[
72x^3yz + 36x^4 + 36x^2y^2z^2 + 18x^3yz = 36x^4 + 90x^3yz + 36x^2y^2z^2
\]

Answer:
\[
\boxed{36x^4 + 90x^3yz + 36x^2y^2z^2}
\]

---

Problem 5:

\[
(-21c^4d^5 - 7d^3)(3c^4d^4 - d^2)
\]

Use the distributive property:
\[
(-21c^4d^5 - 7d^3)(3c^4d^4 - d^2) = (-21c^4d^5)(3c^4d^4) + (-21c^4d^5)(-d^2) + (-7d^3)(3c^4d^4) + (-7d^3)(-d^2)
\]

1. Compute each term:
\[
(-21c^4d^5)(3c^4d^4) = -63c^8d^9
\]
\[
(-21c^4d^5)(-d^2) = 21c^4d^7
\]
\[
(-7d^3)(3c^4d^4) = -21c^4d^7
\]
\[
(-7d^3)(-d^2) = 7d^5
\]

Combine all terms:
\[
-63c^8d^9 + 21c^4d^7 - 21c^4d^7 + 7d^5 = -63c^8d^9 + 7d^5
\]

Answer:
\[
\boxed{-63c^8d^9 + 7d^5}
\]

---

Problem 6:

\[
(m + 10n)(-4n + 2m)
\]

Use the distributive property:
\[
(m + 10n)(-4n + 2m) = (m)(-4n) + (m)(2m) + (10n)(-4n) + (10n)(2m)
\]

1. Compute each term:
\[
(m)(-4n) = -4mn
\]
\[
(m)(2m) = 2m^2
\]
\[
(10n)(-4n) = -40n^2
\]
\[
(10n)(2m) = 20mn
\]

Combine all terms:
\[
2m^2 - 4mn + 20mn - 40n^2 = 2m^2 + 16mn - 40n^2
\]

Answer:
\[
\boxed{2m^2 + 16mn - 40n^2}
\]

---

Problem 7:

\[
(-2gh + 6)\left(-gh + \frac{1}{2}\right)
\]

Use the distributive property:
\[
(-2gh + 6)\left(-gh + \frac{1}{2}\right) = (-2gh)(-gh) + (-2gh)\left(\frac{1}{2}\right) + (6)(-gh) + (6)\left(\frac{1}{2}\right)
\]

1. Compute each term:
\[
(-2gh)(-gh) = 2g^2h^2
\]
\[
(-2gh)\left(\frac{1}{2}\right) = -gh
\]
\[
(6)(-gh) = -6gh
\]
\[
(6)\left(\frac{1}{2}\right) = 3
\]

Combine all terms:
\[
2g^2h^2 - gh - 6gh + 3 = 2g^2h^2 - 7gh + 3
\]

Answer:
\[
\boxed{2g^2h^2 - 7gh + 3}
\]

---

Problem 8:

\[
(-9 + 3p^6)(p - q^4)
\]

Use the distributive property:
\[
(-9 + 3p^6)(p - q^4) = (-9)(p) + (-9)(-q^4) + (3p^6)(p) + (3p^6)(-q^4)
\]

1. Compute each term:
\[
(-9)(p) = -9p
\]
\[
(-9)(-q^4) = 9q^4
\]
\[
(3p^6)(p) = 3p^7
\]
\[
(3p^6)(-q^4) = -3p^6q^4
\]

Combine all terms:
\[
3p^7 - 3p^6q^4 - 9p + 9q^4
\]

Answer:
\[
\boxed{3p^7 - 3p^6q^4 - 9p + 9q^4}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad \frac{1}{9}a^4b^2 + 6a^2b + 81 \\
2. & \quad 121v^2w^2 - 4 \\
3. & \quad -80rs^2 - 20rs + 32tus + 8tu \\
4. & \quad 36x^4 + 90x^3yz + 36x^2y^2z^2 \\
5. & \quad -63c^8d^9 + 7d^5 \\
6. & \quad 2m^2 + 16mn - 40n^2 \\
7. & \quad 2g^2h^2 - 7gh + 3 \\
8. & \quad 3p^7 - 3p^6q^4 - 9p + 9q^4
\end{aligned}
}
\]