Answer key for a math worksheet on multiplying binomials with multi-variable expressions.

It looks like you've uploaded an answer key for a worksheet titled *"Multiplying Binomials"*, which includes 8 problems involving the multiplication of binomials with multiple variables. Since this is an answer key, it already contains the correct solutions.

However, I can help by explaining how to solve each problem step-by-step using the FOIL method (First, Outer, Inner, Last) or distributive property, depending on the structure.

Let’s go through each one:

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

This is a perfect square trinomial: $(A + B)^2 = A^2 + 2AB + B^2$

Let $ A = \frac{1}{3}a^2b $, $ B = 9 $

- $ A^2 = \left(\frac{1}{3}a^2b\right)^2 = \frac{1}{9}a^4b^2 $
- $ 2AB = 2 \cdot \frac{1}{3}a^2b \cdot 9 = 6a^2b $
- $ B^2 = 81 $

✔ Final Answer:
$$
\boxed{\frac{1}{9}a^4b^2 + 6a^2b + 81}
$$

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2) $(11vw - 2)(2 + 11vw)$

Note: This is the same as $(11vw - 2)(11vw + 2)$ — a difference of squares pattern:
$(A - B)(A + B) = A^2 - B^2$

Let $ A = 11vw $, $ B = 2 $

- $ A^2 = (11vw)^2 = 121v^2w^2 $
- $ B^2 = 4 $

So:
$$
(11vw)^2 - (2)^2 = 121v^2w^2 - 4
$$

✔ Final Answer:
$$
\boxed{121v^2w^2 - 4}
$$

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3) $(20rs - 8tu)(-1 - 4s)$

Use distributive property (or FOIL):

Distribute each term in the first binomial to both terms in the second:

- $20rs \cdot (-1) = -20rs$
- $20rs \cdot (-4s) = -80rs^2$
- $-8tu \cdot (-1) = 8tu$
- $-8tu \cdot (-4s) = 32stu$

Now combine all:
$$
-20rs - 80rs^2 + 8tu + 32stu
$$

Reorder terms (usually alphabetical/standard form):
$$
-80rs^2 + 32stu - 20rs + 8tu
$$

✔ Final Answer:
$$
\boxed{-80rs^2 + 32stu - 20rs + 8tu}
$$

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4) $(-12x^3 - 6x^2yz)(-6yz - 3x)$

Distribute each term:

Break it down:

First: $-12x^3 \cdot (-6yz) = 72x^3yz$
Then: $-12x^3 \cdot (-3x) = 36x^4$

Next: $-6x^2yz \cdot (-6yz) = 36x^2y^2z^2$
Then: $-6x^2yz \cdot (-3x) = 18x^3yz$

Now add all terms:
$$
72x^3yz + 36x^4 + 36x^2y^2z^2 + 18x^3yz
$$

Combine like terms: $72x^3yz + 18x^3yz = 90x^3yz$

Final expression:
$$
36x^4 + 90x^3yz + 36x^2y^2z^2
$$

But note: The answer key says: $36x^2y^2z^2 + 90x^3yz + 36x^4$, which is the same.

✔ Final Answer:
$$
\boxed{36x^2y^2z^2 + 90x^3yz + 36x^4}
$$

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5) $(-21c^4d^5 - 7d^3)(3c^4d^4 - d^2)$

Distribute each term:

First: $-21c^4d^5 \cdot 3c^4d^4 = -63c^8d^9$
Then: $-21c^4d^5 \cdot (-d^2) = +21c^4d^7$

Next: $-7d^3 \cdot 3c^4d^4 = -21c^4d^7$
Then: $-7d^3 \cdot (-d^2) = +7d^5$

Now combine:
- $-63c^8d^9$
- $+21c^4d^7 - 21c^4d^7 = 0$
- $+7d^5$

So only two terms remain:
$$
-63c^8d^9 + 7d^5
$$

✔ Final Answer:
$$
\boxed{-63c^8d^9 + 7d^5}
$$

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6) $(m + 10n)(-4n + 2m)$

Rewrite second binomial: $2m - 4n$

Now distribute:

- $m \cdot 2m = 2m^2$
- $m \cdot (-4n) = -4mn$
- $10n \cdot 2m = 20mn$
- $10n \cdot (-4n) = -40n^2$

Add:
- $2m^2$
- $-4mn + 20mn = 16mn$
- $-40n^2$

✔ Final Answer:
$$
\boxed{2m^2 + 16mn - 40n^2}
$$

Wait! But the answer key says: $2m^2 - 40n^2 + 16mn$ — same thing.

Yes, that matches.

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7) $(-2gh + 6)\left(-gh + \frac{1}{2}\right)$

Distribute:

- $-2gh \cdot (-gh) = 2g^2h^2$
- $-2gh \cdot \frac{1}{2} = -gh$
- $6 \cdot (-gh) = -6gh$
- $6 \cdot \frac{1}{2} = 3$

Now combine:
- $2g^2h^2$
- $-gh - 6gh = -7gh$
- $+3$

✔ Final Answer:
$$
\boxed{2g^2h^2 - 7gh + 3}
$$

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8) $(-9 + 3p^6)(p - q^4)$

Distribute:

- $-9 \cdot p = -9p$
- $-9 \cdot (-q^4) = +9q^4$
- $3p^6 \cdot p = 3p^7$
- $3p^6 \cdot (-q^4) = -3p^6q^4$

Now combine:
- $-3p^6q^4$
- $+3p^7$
- $+9q^4$
- $-9p$

Order terms (by degree or alphabetically):

$$
-3p^6q^4 + 3p^7 + 9q^4 - 9p
$$

✔ Final Answer:
$$
\boxed{-3p^6q^4 + 3p^7 + 9q^4 - 9p}
$$

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✔ Summary of All Answers (Matched with Key)

| Problem | Answer |
|--------|--------|
| 1 | $\frac{1}{9}a^4b^2 + 6a^2b + 81$ |
| 2 | $121v^2w^2 - 4$ |
| 3 | $-80rs^2 + 32stu - 20rs + 8tu$ |
| 4 | $36x^2y^2z^2 + 90x^3yz + 36x^4$ |
| 5 | $-63c^8d^9 + 7d^5$ |
| 6 | $2m^2 + 16mn - 40n^2$ |
| 7 | $2g^2h^2 - 7gh + 3$ |
| 8 | $-3p^6q^4 + 3p^7 + 9q^4 - 9p$ |

All match the provided answer key.

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🔍 Tips for Solving These Problems:

1. Always use the distributive property (FOIL works for two binomials).
2. Multiply coefficients and add exponents when multiplying variables.
3. Be careful with signs — negative times negative = positive.
4. Combine like terms at the end.
5. Check your work by rearranging terms in standard order (e.g., highest degree first).

Let me know if you'd like a printable version or practice problems!