To factor the polynomial [tex]\(24x^3 - 54x^2 + 44x - 99\)[/tex] by grouping, let's break down the steps:
1. Group the terms appropriately:
[tex]\[
24x^3 - 54x^2 + 44x - 99
\][/tex]
Group as:
[tex]\[
(24x^3 - 54x^2) + (44x - 99)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- For the first group [tex]\(24x^3 - 54x^2\)[/tex], the GCF is [tex]\(6x^2\)[/tex]:
[tex]\[
24x^3 - 54x^2 = 6x^2(4x - 9)
\][/tex]
- For the second group [tex]\(44x - 99\)[/tex], the GCF is [tex]\(11\)[/tex]:
[tex]\[
44x - 99 = 11(4x - 9)
\][/tex]
3. Notice that after factoring out the GCF from each group, the binomial factors [tex]\(4x - 9\)[/tex] are common:
[tex]\[
24x^3 - 54x^2 + 44x - 99 = 6x^2(4x - 9) + 11(4x - 9)
\][/tex]
4. Factor out the common binomial factor [tex]\((4x - 9)\)[/tex]:
[tex]\[
6x^2(4x - 9) + 11(4x - 9) = (4x - 9)(6x^2 + 11)
\][/tex]
Therefore, the common factor that is missing from both sets of parentheses is:
[tex]\[
6x^2 + 11
\][/tex]
So, the correct answer is:
```
6 x^2 + 11
```
