Let's start by understanding the question and the given expression for the weight of the water bottles, which is [tex]\( 60x^2 + 48x + 24 \)[/tex] ounces.
We're asked to find a factorization that represents the number of water bottles and the weight of each water bottle. Let's look at each factorization option given and see which fits correctly.
### Option 1: [tex]\( 6(10x^2 + 8x + 2) \)[/tex]
Factorizing [tex]\( 60x^2 + 48x + 24 \)[/tex]:
[tex]\[
6 \cdot (10x^2 + 8x + 2) = 60x^2 + 48x + 24
\][/tex]
So, this factorization is correct.
### Option 2: [tex]\( 12(5x^2 + 4x + 2) \)[/tex]
Factorizing:
[tex]\[
12 \cdot (5x^2 + 4x + 2) = 60x^2 + 48x + 24
\][/tex]
So, this factorization is also correct.
### Option 3: [tex]\( 6x(10x^2 + 8x + 2) \)[/tex]
Factorizing:
[tex]\[
6x \cdot (10x^2 + 8x + 2) = 60x^3 + 48x^2 + 12x \][/tex]
This factorization is incorrect since it does not match the given expression [tex]\( 60x^2 + 48x + 24 \)[/tex].
### Option 4: [tex]\( 12x(5x^2 + 4x + 2) \)[/tex]
Factorizing:
[tex]\[
12x \cdot (5x^2 + 4x + 2) = 60x^3 + 48x^2 + 12x
\][/tex]
This factorization is also incorrect since it again does not match the given expression [tex]\( 60x^2 + 48x + 24 \)[/tex].
From the provided options, the correct factorizations that match the given polynomial expression are:
- [tex]\( 6(10x^2 + 8x + 2) \)[/tex]
- [tex]\( 12(5x^2 + 4x + 2) \)[/tex]
Since Mara shared water bottles, the natural interpretation would be connecting the given factorization to the weights and counts directly. The correct factorization representing the number of water bottles and the weight of each are likely:
[tex]\[
6 \left(10x^2 + 8x + 2\right) \quad \text{and} \quad 12 \left(5x^2 + 4x + 2\right)
\][/tex]
If we map the result from the problem check,
- number of water bottles would be 6
- weight of each bottle would be [tex]\(10x^2 + 8x + 2\)[/tex].
Therefore:
[tex]\[ \boxed{6 \left(10x^2 + 8x + 2\right)} \][/tex]
or
- number of water bottles would be 12
- weight of each bottle would be [tex]\(5x^2 + 4x + 2\)[/tex].
Therefore:
[tex]\[ \boxed{12 \left(5x^2 + 4x + 2\right)} \][/tex]
Both factorizations represent potential correct answers.
