Final answer:
To factor the polynomial 12x^3 - 2x^2 + 18x - 3 by grouping, we grouped similar terms and factored out common factors to arrive at (2x^2 + 3)(6x - 1), which most closely resembles option A but is not exactly one of the given choices due to a sign error.
Step-by-step explanation:
To determine the factors of the polynomial 12x3 - 2x2 + 18x - 3 by grouping, we need to split the polynomial into groups that can be factored by common terms or by using the distributive property. Let's analyze the options presented and attempt grouping the terms of the polynomial:
- Grouping the first two and the last two terms: (12x3 - 2x2) + (18x - 3). We can factor out a 2x2 from the first group and a 3 from the second group.
- This results in 2x2(6x - 1) + 3(6x - 1). Now we can see that (6x - 1) is a common factor.
- The polynomial can be factored as (2x2 + 3)(6x - 1).
Reviewing the answer choices, we find that option A, (2x - 1)(6x2 + 4x - 3), is closest to our grouping result. However, it has a slight error as the sign in the first factor should be opposite. Therefore, none of the options perfectly match our factoring by grouping result, but the correct grouping approach does indeed produce a factor similar to that in option A, except for the sign error.