Answer key for expanding algebraic expressions using the distributive property.

It looks like you've uploaded a completed worksheet titled *"Expand the Expression 6:2 Answers"*, which is focused on factoring algebraic expressions using the distributive property. The task asks students to match an expanded expression (like `3a + 15`) with its factored form (like `3(a + 5)`), and the answers are already filled in.

However, since you're asking me to solve the problem and explain the solution, I’ll go through how to solve these types of problems step-by-step, using the examples from the worksheet.

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🔍 Goal:

Use the distributive property in reverse — that is, factor out the greatest common factor (GCF) from each expression to write it in the form:

> $$
> \text{GCF} \times (\text{expression inside parentheses})
> $$

We'll walk through a few examples to show how this works.

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## ✔ Step-by-Step Explanation

Example:

Expression: $ 6x + 12 $

Step 1: Find the GCF of the coefficients.
- Coefficients: 6 and 12
- GCF = 6

Step 2: Factor out the GCF.
- Divide each term by the GCF:
- $ 6x ÷ 6 = x $
- $ 12 ÷ 6 = 2 $
- So: $ 6x + 12 = 6(x + 2) $

✔ Answer: $ 6(x + 2) $

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Now let's do a few from the list:

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🔹 Problem 1: $ 3a + 15 $

- GCF of 3 and 15 is 3
- $ 3a ÷ 3 = a $
- $ 15 ÷ 3 = 5 $
- So: $ 3a + 15 = 3(a + 5) $

✔ Matched correctly!

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🔹 Problem 2: $ 2b + 30 $

- GCF of 2 and 30 is 2
- $ 2b ÷ 2 = b $
- $ 30 ÷ 2 = 15 $
- So: $ 2(b + 15) $

✔ Correct!

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🔹 Problem 3: $ 6c - 18 $

- GCF of 6 and 18 is 6
- $ 6c ÷ 6 = c $
- $ -18 ÷ 6 = -3 $
- So: $ 6(c - 3) $

✔ Correct!

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🔹 Problem 4: $ 5d - 45 $

- GCF of 5 and 45 is 5
- $ 5d ÷ 5 = d $
- $ -45 ÷ 5 = -9 $
- So: $ 5(d - 9) $

✔ Correct!

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🔹 Problem 5: $ 30 - 3e $

- GCF of 30 and 3 is 3
- $ 30 ÷ 3 = 10 $
- $ -3e ÷ 3 = -e $
- So: $ 3(10 - e) $

✔ Correct!

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🔹 Problem 6: $ 28 - 7f $

- GCF of 28 and 7 is 7
- $ 28 ÷ 7 = 4 $
- $ -7f ÷ 7 = -f $
- So: $ 7(4 - f) $

✔ Correct!

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🔹 Problem 7: $ 12g + 20 $

- GCF of 12 and 20 is 4
- $ 12g ÷ 4 = 3g $
- $ 20 ÷ 4 = 5 $
- So: $ 4(3g + 5) $

✔ Correct!

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🔹 Problem 8: $ 14h - 21 $

- GCF of 14 and 21 is 7
- $ 14h ÷ 7 = 2h $
- $ -21 ÷ 7 = -3 $
- So: $ 7(2h - 3) $

✔ Correct!

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🔹 Problem 13: $ 5m + 15n $

- GCF of 5 and 15 is 5
- $ 5m ÷ 5 = m $
- $ 15n ÷ 5 = 3n $
- So: $ 5(m + 3n) $

✔ Correct!

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🔹 Problem 14: $ 10p - 12q $

- GCF of 10 and 12 is 2
- $ 10p ÷ 2 = 5p $
- $ -12q ÷ 2 = -6q $
- So: $ 2(5p - 6q) $

✔ Correct!

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🔹 Problem 20: $ 18y + 36z $

- GCF of 18 and 36 is 18
- $ 18y ÷ 18 = y $
- $ 36z ÷ 18 = 2z $
- So: $ 18(y + 2z) $

✔ Correct!

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## ✔ Summary: How to Solve These Problems

For any expression like $ ax + ay $ or $ ax + b $, follow these steps:

1. Find the Greatest Common Factor (GCF) of all terms.
2. Divide each term by the GCF.
3. Write the expression as:
$$
\text{GCF} \times (\text{sum of results})
$$

> This is the reverse of the distributive property:
> $ a(b + c) = ab + ac $ → reverse: $ ab + ac = a(b + c) $

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## 📌 Final Note:
All the answers on your worksheet are correct! They properly apply the distributive property in reverse to factor the expressions.

If you were supposed to choose among four options, then the correct one would be the one matching the factored form shown here.

Let me know if you'd like a printable version or help with similar problems!