To factor the polynomial [tex]\(x^3 + 6x^2 - 4x - 24\)[/tex] completely, follow these steps:
1. Identify a possible rational root: One method is to use the Rational Root Theorem, which suggests testing possible rational roots that are factors of the constant term ([tex]\(-24\)[/tex]) divided by the leading coefficient (1). Possible rational roots include: [tex]\(\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm24\)[/tex].
2. Test the rational roots: We test [tex]\(x = 2\)[/tex] and find that it is a root because:
[tex]\[
2^3 + 6(2)^2 - 4(2) - 24 = 8 + 24 - 8 - 24 = 0
\][/tex]
3. Factor out [tex]\(x - 2\)[/tex]: Since [tex]\(x = 2\)[/tex] is a root, [tex]\(x - 2\)[/tex] is a factor. We then factor the polynomial by using synthetic division or polynomial division to divide [tex]\(x^3 + 6x^2 - 4x - 24\)[/tex] by [tex]\(x - 2\)[/tex].
Performing the division, we get:
[tex]\[
\frac{x^3 + 6x^2 - 4x - 24}{x - 2} = x^2 + 8x + 12
\][/tex]
4. Factor the quadratic polynomial: Now we need to factor [tex]\(x^2 + 8x + 12\)[/tex]. We look for two numbers that multiply to 12 and add up to 8. These numbers are 6 and 2. Therefore,
[tex]\[
x^2 + 8x + 12 = (x + 2)(x + 6)
\][/tex]
5. Combine the factors: Now we combine all the factors found:
[tex]\[
x^3 + 6x^2 - 4x - 24 = (x - 2)(x + 2)(x + 6)
\][/tex]
Therefore, the polynomial [tex]\(x^3 + 6x^2 - 4x - 24\)[/tex] factors completely as:
[tex]\[
\boxed{(x - 2)(x + 2)(x + 6)}
\][/tex]
Among the given options, this corresponds to
[tex]\[
(x+2)(x-2)(x+6)
\][/tex]
