Question

mara carried water bottles to the field to share with her team at halftime. the water bottles weighed a total of $60x^2 + 48x + 24$ ounces.
which factorization could represent the number of water bottles and weight of each water bottle?
\\(\bigcirc\\) $6(10x^2 + 8x + 2)$
\\(\bigcirc\\) $12(5x^2 + 4x + 2)$
\\(\bigcirc\\) $6x(10x^2 + 8x + 2)$
\\(\bigcirc\\) $12x(5x^2 + 4x + 2)$

Explanation:

Step1: Find GCF of coefficients

The coefficients are 60, 48, 24. GCF of 60, 48, 24 is 12? Wait, no: 60 ÷ 12 = 5, 48 ÷ 12 = 4, 24 ÷ 12 = 2? Wait, no, 60, 48, 24: GCF of 60 and 48 is 12? Wait 60 = 12×5, 48=12×4, 24=12×2? Wait no, 24×2=48, 24×2.5=60? No, 60, 48, 24: let's factor each:

60: 2×2×3×5

48: 2×2×2×2×3

24: 2×2×2×3

So common factors: 2×2×3 = 12? Wait no, 2×2×3 is 12? Wait 2×2×3=12, but 24 ÷12=2, 48÷12=4, 60÷12=5. Wait but wait, the original polynomial is \(60x^2 + 48x + 24\). Let's check the options. Let's factor out the GCF.

Wait, let's check each option by expanding:

Option 1: \(6(10x^2 + 8x + 2) = 60x^2 + 48x + 12\) → Not equal to original (24 vs 12). So wrong.

Option 2: \(12(5x^2 + 4x + 2) = 60x^2 + 48x + 24\) → Correct expansion.

Option 3: \(6x(10x^2 + 8x + 2) = 60x^3 + 48x^2 + 12x\) → Degree 3, original is degree 2. Wrong.

Option 4: \(12x(5x^2 + 4x + 2) = 60x^3 + 48x^2 + 24x\) → Degree 3, wrong.

Wait, wait I made a mistake earlier. Let's recheck Option 1: 6(10x² +8x +2) = 60x² +48x +12. But original is +24. So that's wrong. Option 2: 12(5x² +4x +2) = 60x² +48x +24. Which matches. So the GCF is 12? Wait but earlier when I factored 60,48,24, the GCF is 12? Wait 60,48,24: GCF is 12? Wait 24 is the constant term, 48 is 24×2, 60 is 24×2.5? No, 24×2=48, 24×2.5=60? No, 60=24×2.5 is not integer. Wait, no, 60, 48, 24: the GCF is 12? Wait 24 is 12×2, 48 is 12×4, 60 is 12×5. Yes, 12 is a common factor. Wait but when I factored 6*(10x²+8x+2), I got 60x²+48x+12, which is missing 12. So Option 2 is correct.

Step2: Verify other options

Option 3 and 4 have x terms, which would make the polynomial have x in all terms, but original has \(x^2\), \(x\), and constant. So factoring out x would introduce an x in the constant term, which is not present. So options 3 and 4 are out. Option 1 gives constant term 12, original is 24. So Option 2 is correct.

Answer:

B. \(12(5x^2 + 4x + 2)\) (Wait, the options are labeled with circles, but in the problem, the second option is "12(5x² + 4x + 2)". So the correct option is the second one, which is "12(5x² + 4x + 2)".