Question

The area of the rectangle is given. Factor out the GCF to find expressions for the missing dimensions.
21 a cubed b Superscript 4 Baseline plus 14 a squared b cubed plus 28 a Superscript 4 Baseline b squared
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Part 1
Find expressions for the missing dimensions.
  
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​(Simplify your answer. Use a comma to separate answers as​ needed.)

Answer 1

To factor out the greatest common factor (GCF) from the expression \( 21a^3b^4 + 14a^2b^3 + 28a^4b^2 \), we first need to identify the GCF of the coefficients and the variables.

### Step 1: Find the GCF of the coefficients
- The coefficients are \( 21, 14, \) and \( 28 \).
- The GCF of these numbers is \( 7 \).

### Step 2: Find the GCF of the variable parts
- For \( a \), the smallest exponent among \( a^3, a^2, \) and \( a^4 \) is \( a^2 \).
- For \( b \), the smallest exponent among \( b^4, b^3, \) and \( b^2 \) is \( b^2 \).

### Step 3: Combine the GCF
Now we can combine the GCF of the coefficients and the variables:

\[
\text{GCF} = 7a^2b^2
\]

### Step 4: Factor out the GCF
Now, we will factor \( 7a^2b^2 \) out of the expression:

1. Divide each term by \( 7a^2b^2 \):
- \( 21a^3b^4 \div 7a^2b^2 = 3a^{3-2}b^{4-2} = 3ab^2 \)
- \( 14a^2b^3 \div 7a^2b^2 = 2b^{3-2} = 2b \)
- \( 28a^4b^2 \div 7a^2b^2 = 4a^{4-2} = 4a^2 \)

### Step 5: Write the factored expression
Putting it all together, we have:

\[
21a^3b^4 + 14a^2b^3 + 28a^4b^2 = 7a^2b^2(3ab^2 + 2b + 4a^2)
\]

### Conclusion
The expressions for the missing dimensions in terms of the GCF factored out are:

**Answers:**

**7a²b², 3ab² + 2b + 4a²**