How can I determine the factors of the polynomial x³ + 4x² + 5x + 20 by grouping?

To factor the polynomial x³ + 4x² + 5x + 20 by grouping, follow these steps:

1. Group terms: Divide the polynomial into two groups. For this polynomial, we can group the first two terms and the last two terms:

   (x³ + 4x²) + (5x + 20)

2. Factor out common factors:
   • From the first group (x³ + 4x²), we can factor out x²:

     x²(x + 4)

   • From the second group (5x + 20), we can factor out 5:

     5(x + 4)

3. Combine the factored groups: Now, substitute the factored results back into the polynomial:

   x²(x + 4) + 5(x + 4)

4. Factor out the common binomial: Both groups contain the common factor (x + 4), so we can factor this out:

   (x + 4)(x² + 5)

5. Final result: The factored form of the polynomial x³ + 4x² + 5x + 20 is:

   (x + 4)(x² + 5)

This shows how you can systematically approach factoring by grouping. It’s an effective method especially when the polynomial contains terms that can be grouped conveniently.