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------------------------------------------------ Factor the following expression:

[tex]\[ 6x^3 - 36x^2 - 8x + 48 \][/tex]

Sure! Let's work through factoring the expression step by step:

We need to factor the expression [tex]$6x^3 - 36x^2 - 8x + 48$[/tex].

1. Look for the Greatest Common Factor (GCF):
First, we look for the greatest common factor among all the terms. Here, each coefficient (6, -36, -8, and 48) is divisible by 2. Factoring out the GCF of 2, we get:
[tex]\[
6x^3 - 36x^2 - 8x + 48 = 2(3x^3 - 18x^2 - 4x + 24)
\][/tex]

2. Factor by Grouping:
Now, we focus on factoring what is inside the parentheses. We can try factoring by grouping:

- Group the terms: [tex]$(3x^3 - 18x^2)$[/tex] and [tex]$(-4x + 24)$[/tex].

- Factor out the greatest common factor from each group:
- From [tex]$3x^3 - 18x^2$[/tex], factor out [tex]$3x^2$[/tex]:
[tex]\[
3x^2(x - 6)
\][/tex]
- From [tex]$-4x + 24$[/tex], factor out [tex]$-4$[/tex]:
[tex]\[
-4(x - 6)
\][/tex]

- We then write:
[tex]\[
3x^2(x - 6) - 4(x - 6)
\][/tex]

3. Factor out the common binomial:
Both terms now contain the common factor [tex]$(x - 6)$[/tex]. We factor [tex]$(x - 6)$[/tex] out:
[tex]\[
(3x^2 - 4)(x - 6)
\][/tex]

4. Combine everything:
Don't forget the factor of 2 we took out at the beginning:
[tex]\[
2(3x^2 - 4)(x - 6)
\][/tex]

Thus, the factored form of the expression [tex]$6x^3 - 36x^2 - 8x + 48$[/tex] is [tex]$\boxed{2(x - 6)(3x^2 - 4)}$[/tex].