To determine which expression is equal to [tex]\(-x^2 - 36\)[/tex], we need to expand each option and compare the results.
### Option A: [tex]\((-x - 6i)(x + 6i)\)[/tex]
We can use the distributive property (FOIL method) to expand this:
[tex]\[
(-x - 6i)(x + 6i) = (-x)(x) + (-x)(6i) + (-6i)(x) + (-6i)(6i)
\][/tex]
Now we calculate each term:
[tex]\[
(-x)(x) = -x^2
\][/tex]
[tex]\[
(-x)(6i) = -6xi
\][/tex]
[tex]\[
(-6i)(x) = -6xi
\][/tex]
[tex]\[
(-6i)(6i) = -36(-1) = 36 \quad \text{(because } i^2 = -1)
\][/tex]
Combining all terms, we get:
[tex]\[
-x^2 - 6xi - 6xi + 36 = -x^2 - 12xi + 36
\][/tex]
This is not equal to [tex]\(-x^2 - 36\)[/tex].
### Option B: [tex]\((x + 6i)(x - 6i)\)[/tex]
Use the difference of squares formula:
[tex]\[
(x + 6i)(x - 6i) = x^2 - (6i)^2
\][/tex]
Since [tex]\((6i)^2 = 36(-1) = -36\)[/tex], we get:
[tex]\[
x^2 - (-36) = x^2 + 36
\][/tex]
This is not equal to [tex]\(-x^2 - 36\)[/tex].
### Option C: [tex]\((-x + 6i)(x - 6i)\)[/tex]
Again, we use the distributive property to expand:
[tex]\[
(-x + 6i)(x - 6i) = (-x)(x) + (-x)(-6i) + (6i)(x) + (6i)(-6i)
\][/tex]
Now we calculate each term:
[tex]\[
(-x)(x) = -x^2
\][/tex]
[tex]\[
(-x)(-6i) = 6xi
\][/tex]
[tex]\[
(6i)(x) = 6xi
\][/tex]
[tex]\[
(6i)(-6i) = -36 \quad \text{(because } i^2 = -1)
\][/tex]
Combining all terms, we get:
[tex]\[
-x^2 + 6xi + 6xi - 36 = -x^2 + 12xi - 36
\][/tex]
This is not equal to [tex]\(-x^2 - 36\)[/tex].
### Option D: [tex]\((-x - 6i)(x - 6i)\)[/tex]
Use the distributive property:
[tex]\[
(-x - 6i)(x - 6i) = (-x)(x) + (-x)(-6i) + (-6i)(x) + (-6i)(-6i)
\][/tex]
Now we calculate each term:
[tex]\[
(-x)(x) = -x^2
\][/tex]
[tex]\[
(-x)(-6i) = 6xi
\][/tex]
[tex]\[
(-6i)(x) = -6xi
\][/tex]
[tex]\[
(-6i)(-6i) = -36 \quad \text{(because } i^2 = -1)
\][/tex]
Combining all terms, we get:
[tex]\[
-x^2 + 6xi - 6xi - 36 = -x^2 - 36
\][/tex]
This matches the given expression [tex]\(-x^2 - 36\)[/tex].
### Conclusion:
The correct option is D: [tex]\((-x - 6i)(x - 6i)\)[/tex]
