How do you factor the polynomial 4x^4 + 20x^2 + 3x^2 + 15 by grouping, and what is the resulting expression?

To factor the polynomial 4x⁴ + 20x² + 3x² + 15 by grouping, we can follow these steps:

1. First, let’s rearrange the polynomial to group the terms effectively:
2. (4x⁴ + 20x²) + (3x² + 15)
3. Next, factor out the common factors in each group:
• In the first group (4x⁴ + 20x²), we can factor out 4x²:
• 4x²(x² + 5)
• In the second group (3x² + 15), we can factor out 3:
• 3(x² + 5)
4. Now we can rewrite the polynomial as:
5. 4x²(x² + 5) + 3(x² + 5)
6. Notice that we have a common factor of (x² + 5) in both terms. We can factor this out:
7. So we get:
8. (x² + 5)(4x² + 3)

Thus, the resulting expression after factoring the polynomial 4x⁴ + 20x² + 3x² + 15 by grouping is:

(x² + 5)(4x² + 3)