To determine which expression is equivalent to [tex]\(-x^2 - 81\)[/tex], we need to simplify each of the given expressions and compare the results.
### Evaluation of Each Choice
Choice A: [tex]\((-x - 9i)(x + 9i)\)[/tex]
First, apply the distributive property (FOIL method):
[tex]\[
(-x - 9i)(x + 9i) = (-x)(x) + (-x)(9i) + (-9i)(x) + (-9i)(9i)
\][/tex]
Simplify each term:
[tex]\[
(-x)(x) = -x^2, \quad (-x)(9i) = -9xi, \quad (-9i)(x) = -9xi, \quad (-9i)(9i) = -81i^2
\][/tex]
Combine like terms and use the fact that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
-x^2 - 9xi - 9xi - 81i^2 = -x^2 - 18xi - 81(-1) = -x^2 - 18xi + 81
\][/tex]
Therefore:
[tex]\[
(-x - 9i)(x + 9i) = -x^2 + 81 \quad \Rightarrow \quad \text{not equal to } -x^2 - 81
\][/tex]
Choice B: [tex]\((-x - 9i)(x - 9i)\)[/tex]
First, apply the distributive property (FOIL method):
[tex]\[
(-x - 9i)(x - 9i) = (-x)(x) + (-x)(-9i) + (-9i)(x) + (-9i)(-9i)
\][/tex]
Simplify each term:
[tex]\[
(-x)(x) = -x^2, \quad (-x)(-9i) = 9xi, \quad (-9i)(x) = -9xi, \quad (-9i)(-9i) = 81i^2
\][/tex]
Combine like terms and use the fact that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
-x^2 + 9xi - 9xi + 81i^2 = -x^2 + 81(-1) = -x^2 - 81
\][/tex]
Therefore:
[tex]\[
(-x - 9i)(x - 9i) = -x^2 - 81 \quad \Rightarrow \quad \text{equal to } -x^2 - 81
\][/tex]
Choice C: [tex]\((x + 9i)(x - 9i)\)[/tex]
First, apply the distributive property (FOIL method):
[tex]\[
(x + 9i)(x - 9i) = (x)(x) + (x)(-9i) + (9i)(x) + (9i)(-9i)
\][/tex]
Simplify each term:
[tex]\[
(x)(x) = x^2, \quad (x)(-9i) = -9xi, \quad (9i)(x) = 9xi, \quad (9i)(-9i) = -81i^2
\][/tex]
Combine like terms and use the fact that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
x^2 - 9xi + 9xi - 81i^2 = x^2 - 81(-1) = x^2 + 81
\][/tex]
Therefore:
[tex]\[
(x + 9i)(x - 9i) = x^2 + 81 \quad \Rightarrow \quad \text{not equal to } -x^2 - 81
\][/tex]
Choice D: [tex]\((-x + 9i)(x - 9i)\)[/tex]
First, apply the distributive property (FOIL method):
[tex]\[
(-x + 9i)(x - 9i) = (-x)(x) + (-x)(-9i) + (9i)(x) + (9i)(-9i)
\][/tex]
Simplify each term:
[tex]\[
(-x)(x) = -x^2, \quad (-x)(-9i) = 9xi, \quad (9i)(x) = 9xi, \quad (9i)(-9i) = -81i^2
\][/tex]
Combine like terms and use the fact that [tex]\(i^2 = -1\)[/tex]:
[tex]\[
-x^2 + 9xi + 9xi - 81i^2 = -x^2 + 18xi - 81(-1) = -x^2 + 18xi + 81
\][/tex]
Therefore:
[tex]\[
(-x + 9i)(x - 9i) = -x^2 + 81 \quad \Rightarrow \quad \text{not equal to } -x^2 - 81
\][/tex]
### Conclusion
After simplifying all the given choices, we see that only Choice B: [tex]\((-x - 9i)(x - 9i)\)[/tex] simplifies to [tex]\(-x^2 - 81\)[/tex].
Thus, the correct answer is:
[tex]\(\boxed{B}\)[/tex]
