Final answer:
The factored form of the polynomial 4x³ – 8x² – 9x + 18 is option C. (2x + 3)(2x - 3)(x - 2), which is found by recognizing patterns and testing potential factors of the polynomial.
Step-by-step explanation:
The factored form of the expression 4x³ – 8x² – 9x + 18 is found by grouping and factoring by pairs or using trial and error with the possible rational roots, given by the factors of the constant term over the factors of the leading coefficient. The correct factored form must produce the original polynomial when multiplied out. The possible pairs for factoring could be (4x³ - 8x²) + (-9x + 18) or other combinations, but considering the factors of -9 and 18 could lead us to try the factors ± 1, ± 2, ± 3, ± 6, ± 9, and ± 18 against the leading coefficient 4, with factors ± 1, ± 2, and ± 4. By testing these options, we find that (2x + 3)(2x - 3)(x - 2) multiplies to give the original polynomial.
The correct factored form is therefore: (2x + 3)(2x - 3)(x - 2).
Answer choices A and D can be eliminated because they do not exist: '2.1' is not a valid expression within polynomial factoring, while B and C can be analyzed further. The difference of squares in C suggests it is a plausible factorization, particularly since (2x + 3) and (2x - 3) multiply to 4x²-9, a pattern seen in the original polynomial. After fully evaluating all options, it's clear that the correct answer is option C.