Let's analyze the given mathematical expression step-by-step to determine which of the provided options is a factor of the expression:
[tex]\[ 6z^4 - 4 + 9(y^3 + 3) \][/tex]
First, we start by simplifying the expression and examining each part:
1. Identify parts within parentheses:
[tex]\[
9(y^3 + 3) = 9y^3 + 27
\][/tex]
2. Rewriting the given expression with the simplified part:
[tex]\[
6z^4 - 4 + 9y^3 + 27
\][/tex]
3. Combine like terms (if any are present):
[tex]\[
6z^4 + 9y^3 - 4 + 27
\][/tex]
4. Simplify the constants:
[tex]\[
-4 + 27 = 23
\][/tex]
So, the expression becomes:
[tex]\[
6z^4 + 9y^3 + 23
\][/tex]
Now let's review the options to see if any of them can be considered a factor of the original expression:
- Option A: [tex]\( 9(y^3 + 3) = 9y^3 + 27 \)[/tex]
- Option B:
[tex]\[
6z^4 - 4
\][/tex]
- Option C:
[tex]\[
(y^3 + 3)
\][/tex]
- Option D:
[tex]\[
-4 + 9(y^3 + 3) = -4 + 9y^3 + 27 = 9y^3 + 23
\][/tex]
To determine which is a factor, we compare the complete forms to our expression [tex]\( 6z^4 + 9y^3 + 23 \)[/tex]:
- Option A provides the segment [tex]\( 9y^3 + 27 \)[/tex], which when slotted into the full expression appears in part.
- Option B provides [tex]\( 6z^4 - 4 \)[/tex], another segment found in the complete expression.
- Comparing Option C we note it is exactly in the expansion of Option A but independently is not the same as the expression layout.
- Option D is analogous in makeup to elements seen, cut form suggesting split forms but look into organizing from original manner
So, based upon more than direct alignment its still problem directly concurring and factor equivalence is seen strongest in section:
\[
distinct part visible contained for portion factor supporters
Answer:
```
A.\( 9(y^3 + 3)) such order alignment systemic overall factor \ thus constituting particularly within solution form.
