Answer :

To factor the polynomial [tex]\(18x^3 - 27x^2 - 32x + 48\)[/tex] completely, follow these steps:

1. Group the terms:
We start by grouping the polynomial into two pairs to make it easier to factor by grouping.

[tex]\[
(18x^3 - 27x^2) - (32x - 48)
\][/tex]

2. Factor out the greatest common factor (GCF) from each pair:
- In the first pair [tex]\(18x^3 - 27x^2\)[/tex], the GCF is [tex]\(9x^2\)[/tex]. Factoring it out gives us:
[tex]\[
9x^2(2x - 3)
\][/tex]
- In the second pair [tex]\(-32x + 48\)[/tex], the GCF is [tex]\(-16\)[/tex]. Factoring it out gives us:
[tex]\[
-16(2x - 3)
\][/tex]

3. Factor out the common binomial:
Now, we notice that both terms have a common factor of [tex]\((2x - 3)\)[/tex].
We factor [tex]\((2x - 3)\)[/tex] out:
[tex]\[
(2x - 3)(9x^2 - 16)
\][/tex]

4. Recognize and factor the difference of squares:
The expression inside the second set of parentheses, [tex]\(9x^2 - 16\)[/tex], is a difference of squares:
[tex]\[
9x^2 - 16 = (3x)^2 - 4^2 = (3x - 4)(3x + 4)
\][/tex]

5. Combine all the factors:
So, the completely factored form of the polynomial is:
[tex]\[
(2x - 3)(3x - 4)(3x + 4)
\][/tex]

This gives us the fully factored expression: [tex]\((2x - 3)(3x - 4)(3x + 4)\)[/tex].

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