To determine which expression is equivalent to [tex]\(-x^2 - 81\)[/tex], we'll simplify each given choice and compare the results to [tex]\(-x^2 - 81\)[/tex].
First, let’s recall the factorizations of complex expressions. The general form of the difference of squares is:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
Given [tex]\(-x^2 - 81\)[/tex], we can rearrange it as:
[tex]\[ -x^2 - 81 = - (x^2 + 81) \][/tex]
Now the term inside the parenthesis is [tex]\(x^2 + 81\)[/tex] which is a sum of squares and can be factored into:
[tex]\[ x^2 + 81 = x^2 + (9i)^2 \][/tex]
[tex]\[ x^2 + 81 = (x + 9i)(x - 9i) \][/tex]
Thus:
[tex]\[ - (x^2 + 81) = - (x + 9i)(x - 9i) \][/tex]
Our target expression is thus:
[tex]\[ -x^2 - 81 = - (x + 9i)(x - 9i) \][/tex]
Let’s evaluate each given choice to see if it simplifies to [tex]\(- (x + 9i)(x - 9i)\)[/tex]:
### Choice A: [tex]\((-x - 9i)(x + 9i)\)[/tex]
[tex]\[ (-x - 9i)(x + 9i) = -x(x + 9i) - 9i(x + 9i) \][/tex]
[tex]\[ = -x^2 - 9ix - 9ix - 81i^2 \][/tex]
[tex]\[ = -x^2 - 18ix - 81(-1) \][/tex]
[tex]\[ = -x^2 - 18ix + 81 \][/tex]
This is not equal to [tex]\(-x^2 - 81\)[/tex].
### Choice B: [tex]\((-x - 9i)(x - 9i)\)[/tex]
[tex]\[ (-x - 9i)(x - 9i) = -x(x - 9i) - 9i(x - 9i) \][/tex]
[tex]\[ = -x^2 + 9ix - 9ix + 81i^2 \][/tex]
[tex]\[ = -x^2 + 81i^2 \][/tex]
[tex]\[ = -x^2 + 81(-1) \][/tex]
[tex]\[ = -x^2 - 81 \][/tex]
This matches [tex]\(-x^2 - 81\)[/tex].
### Choice C: [tex]\((x + 9i)(x - 9i)\)[/tex]
[tex]\[ (x + 9i)(x - 9i) = x^2 - (9i)^2 \][/tex]
[tex]\[ = x^2 - 81(-1) \][/tex]
[tex]\[ = x^2 + 81 \][/tex]
This does not match [tex]\(-x^2 - 81\)[/tex].
### Choice D: [tex]\((-x + 9i)(x - 9i)\)[/tex]
[tex]\[ (-x + 9i)(x - 9i) = -x(x - 9i) + 9i(x - 9i) \][/tex]
[tex]\[ = -x^2 + 9ix - 9ix + 81i^2 \][/tex]
[tex]\[ = -x^2 + 81i^2 \][/tex]
[tex]\[ = -x^2 + 81(-1) \][/tex]
[tex]\[ = -x^2 - 81 \][/tex]
This also matches [tex]\(-x^2 - 81\)[/tex].
Hence, the correct choices are B and D. Since the question appears to be asking for a single expression, and typically such multiple-choice questions intend one distinct answer (potential oversight if it's a single-selection setting),
Choice B is the most fitting and correct selection.
