Which Shows One Way to Determine the Factors of x³ + 12x² + 2x + 24 by Grouping?

To factor the polynomial x³ + 12x² + 2x + 24 by grouping, we can follow these steps:

1. First, we group the terms in pairs: (x³ + 12x²) and (2x + 24).
2. Next, we factor out the greatest common factor (GCF) from each group:
• From the first group, x² is the GCF, so we factor it out:
• x²(x + 12)
• From the second group, 2 is the GCF, so we factor it out:
• 2(x + 12)
3. Now our expression looks like this:
• x²(x + 12) + 2(x + 12)
4. Notice that (x + 12) is a common factor in both terms, so we can factor that out:
• (x + 12)(x² + 2)
5. Thus, the complete factorization of the polynomial x³ + 12x² + 2x + 24 is:
• (x + 12)(x² + 2)

In conclusion, we used the method of grouping to successfully find the factors of the given polynomial. Factoring allows us to rewrite the polynomial in a product form, which can be useful for solving equations or analyzing the behavior of the function.