Final answer:
The factored form of the given polynomial, after simplifying and combining like terms, and factoring by grouping, seems to be 2(45x²(x + 4) + 2(x + 2)). Without additional context, this is the apparent factored result.
Step-by-step explanation:
The student asked for the completely factored form of the polynomial 5x³ + 20x² + 2x + 85x²(x + 4) + 2(x + 4). To factor this polynomial, we must first expand the terms and then combine like terms. The term 85x²(x + 4) would expand to 85x³ + 340x² and the term 2(x + 4) would expand to 2x + 8. After expanding, we then combine like terms:
- 5x³ + 85x³ = 90x³
- 20x² + 340x² = 360x²
- 2x + 2x = 4x
Therefore, the polynomial can be rewritten as:
90x³ + 360x² + 4x + 8
The next step is to factor by grouping. We can factor out common terms from pairs of these terms:
- 90x³ + 360x² can be factored as 90x²(x + 4)
- 4x + 8 can be factored as 4(x + 2)
Now the polynomial is:
90x²(x + 4) + 4(x + 2)
Finally, we can factor out the common term of 2 from the entire polynomial:
2(45x²(x + 4) + 2(x + 2))
Without further information, this appears to be the factored form under common factoring techniques. To ensure completeness, one would generally check if the quadratic terms can be factored further, but with the given expanded polynomial, this is the current factored result.