To determine which of the given options is a factor of the expression
[tex]\[ 7z^4 - 5 + 10(y^3 + 2), \][/tex]
we need to explore each option and see if it can evenly divide the entire expression.
Option A: [tex]\(10(y^3 + 2)\)[/tex]
Let's rewrite the expression by separating the term [tex]\(10(y^3 + 2)\)[/tex] from the rest:
[tex]\[ 7z^4 - 5 + 10(y^3 + 2) = 7z^4 - 5 + 10y^3 + 20. \][/tex]
Here, [tex]\(10(y^3 + 2)\)[/tex] is clearly one of the summands of the original expression. Hence, we see it's not a factor of the entire expression since in order to be a factor, it must divide the entire expression.
Option B: [tex]\(y^3 + 2\)[/tex]
Again, let's look at the original expression and see if [tex]\((y^3 + 2)\)[/tex] is a factor. If we isolate the term involving [tex]\((y^3 + 2)\)[/tex], we get:
[tex]\[ 7z^4 - 5 + 10(y^3 + 2). \][/tex]
Factoring [tex]\(10\)[/tex] out of [tex]\(10(y^3 + 2)\)[/tex], we get:
[tex]\[ 7z^4 - 5 + 10y^3 + 20. \][/tex]
[tex]\(y^3 + 2\)[/tex] does not cleanly divide [tex]\(7z^4 - 5 + 10y^3 + 20\)[/tex] as a whole, so it's not a factor.
Option C: [tex]\(7z^4 - 5\)[/tex]
Evaluate this part of the expression:
[tex]\[ 7z^4 - 5 + 10(y^3 + 2) = 7z^4 - 5 + 10y^3 + 20. \][/tex]
The term [tex]\(7z^4 - 5\)[/tex] appears to be a distinct part of the original expression but does not divide the entire expression as a whole. Hence, it's not a factor.
Option D: [tex]\(-5 + 10(y^3 + 2)\)[/tex]
Let's simplify the term:
[tex]\[ -5 + 10(y^3 + 2). \][/tex]
Notice that this term occurs without the [tex]\(7z^4\)[/tex]:
When simplified:
[tex]\[ -5 + 10(y^3 + 2) = -5 + 10y^3 + 20 = 10y^3 + 15. \][/tex]
However, looking at the original expression, [tex]\(-5 + 10(y^3 + 2)\)[/tex] should be re-binded correctly to give [tex]\(+15\)[/tex], implying that [tex]\(-5 + 10(y^3 + 2)\)[/tex] is indeed a factor of the entire expression as it matches a part of it correctly.
Thus, the correct answer is:
D. [tex]\(-5 + 10(y^3 + 2)\)[/tex]
