To determine which of the given factorizations can represent Logan's monthly savings pattern for the polynomial [tex]\(300m^2 + 120m + 180\)[/tex], we can follow these steps:
1. Factor the polynomial [tex]\(300m^2 + 120m + 180\)[/tex].
2. Check if any of the given options match this factorization.
Let's start with the polynomial [tex]\(300m^2 + 120m + 180\)[/tex]. Notice anything common in all the terms? We see that each term is divisible by 60.
Factor out 60 from the polynomial:
[tex]\[ 300m^2 + 120m + 180 = 60(5m^2 + 2m + 3) \][/tex]
Now we have the factored form of the polynomial: [tex]\(60(5m^2 + 2m + 3)\)[/tex].
Next, let's compare this with the given options:
1. [tex]\(10m(30m^2 + 12m + 18)\)[/tex]
2. [tex]\(30(10m^2 + 4m + 60)\)[/tex]
3. [tex]\(60(5m^2 + 2m + 3)\)[/tex]
By inspection, we see that the factored form [tex]\(60(5m^2 + 2m + 3)\)[/tex] exactly matches the third option.
Thus, the correct factorization that represents Logan's monthly savings is:
[tex]\[ 60 \left( 5m^2 + 2m + 3 \right) \][/tex]
This indicates that Logan saves an amount represented by the polynomial [tex]\(5m^2 + 2m + 3\)[/tex] each month for 60 months.
