Final answer:
Option A. The polynomial that can be factored into the binomial (x - 6) and the trinomial (-2x^2 + x + 9) is option A: -2x^3 + 13x^2 + 3x - 54. This is determined by performing the polynomial multiplication of the given binomial and trinomial.
Step-by-step explanation:
The student is asking which of the listed polynomials can be factored into the binomial (x – 6) and the trinomial (–2x^2 + x + 9). To determine this, we can perform polynomial multiplication between the binomial and the trinomial to find which option matches the resulting product.
Multiplying (x – 6) by (–2x^2 + x + 9) gives:
- x * (–2x^2) = –2x^3
- x * x = x^2
- x * 9 = 9x
- (–6) * (–2x^2) = 12x^2
- (–6) * x = –6x
- (–6) * 9 = –54
Combining like terms, we get the polynomial –2x^3 + 13x^2 + 3x – 54, which matches option A.