What is the result of factoring the expression 15x³ + 5x² + 6x + 2 by grouping?

To factor the polynomial expression 15x³ + 5x² + 6x + 2 by grouping, we will follow a systematic approach. Here’s how we do it:

1. First, let’s group the terms:

(15x³ + 5x²) + (6x + 2)

2. Next, we factor out the common factors from each group:

From the first group (15x³ + 5x²), we can factor out 5x²:
5x²(3x + 1)

From the second group (6x + 2), we can factor out 2:
2(3x + 1)

3. Now we can rewrite the expression using these factored forms:

5x²(3x + 1) + 2(3x + 1)

Notice that (3x + 1) is a common factor in both terms. We can factor out (3x + 1):
(3x + 1)(5x² + 2)

4. In conclusion, the factored form of the expression 15x³ + 5x² + 6x + 2 by grouping is:

(3x + 1)(5x² + 2)

This means the resulting expression after factoring by grouping is (3x + 1)(5x² + 2).