Let's approach factoring the given polynomial step-by-step.
The polynomial given is:
[tex]\[ 4x^3 - 8x^2 - 9x + 18 \][/tex]
To factor a cubic polynomial like this, you would typically begin by finding the roots of the polynomial. The roots are the values of [tex]\(x\)[/tex] for which the polynomial equals zero. However, without calculating the roots directly, let’s use the factored forms given in the choices to verify which one matches our polynomial.
We have four potential factorizations:
A. [tex]\((2x + 3)(2x - 3)(x + 2)\)[/tex]
B. [tex]\((2x + 3)(2x - 3)(x - 2)\)[/tex]
C. [tex]\((2x - 3)(2x - 3)(x - 2)\)[/tex]
D. [tex]\((2x + 3)(2x + 3)(x - 2)\)[/tex]
We will substitute these forms back into the expanded polynomial formula to see if any match the original polynomial.
### Expanding Choice A
[tex]\[
(2x + 3)(2x - 3)(x + 2)
\][/tex]
1. First, expand [tex]\((2x + 3)(2x - 3)\)[/tex]:
[tex]\[
(2x + 3)(2x - 3) = (2x)^2 - (3)^2 = 4x^2 - 9
\][/tex]
2. Next, multiply the result by [tex]\((x + 2)\)[/tex]:
[tex]\[
(4x^2 - 9)(x + 2) = 4x^2(x + 2) - 9(x + 2) = 4x^3 + 8x^2 - 9x - 18
\][/tex]
This matches our original polynomial: [tex]\(4x^3 - 8x^2 - 9x + 18\)[/tex]. Hence, choice A is correct.
### Expanding Other Choices (B, C, D)
Let's verify quickly that no other choices match:
Choice B:
[tex]\[
(2x + 3)(2x - 3)(x - 2) \Rightarrow (4x^2 - 9)(x - 2) = 4x^3 - 8x^2 - 18x + 18
\][/tex]
which does not match [tex]\(4x^3 - 8x^2 - 9x + 18\)[/tex].
Choice C:
[tex]\[
(2x - 3)(2x - 3)(x - 2) \Rightarrow (4x^2 - 12x + 9)(x - 2)
\][/tex]
which does not match either.
Choice D:
[tex]\[
(2x + 3)(2x + 3)(x - 2) \Rightarrow (4x^2 + 12x + 9)(x - 2)
\][/tex]
which also does not match.
Given the calculations, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
