Let's break down the given expression step-by-step and simplify it. The given expression is:
[tex]\[
(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})
\][/tex]
Step 1: Simplify each square root component.
[tex]\[
\sqrt{9} = 3
\][/tex]
[tex]\[
\sqrt{-4} = 2i \, (\text{since} \, \sqrt{-1} = i \, \text{and} \, \sqrt{4} = 2)
\][/tex]
[tex]\[
2 \sqrt{576} = 2 \times 24 = 48
\][/tex]
[tex]\[
\sqrt{-64} = 8i \, (\text{since} \, \sqrt{-1} = i \, \text{and} \, \sqrt{64} = 8)
\][/tex]
Step 2: Substitute the simplified components back into the expression.
[tex]\[
(-3 + 2i) - (48 + 8i)
\][/tex]
Step 3: Simplify further.
Combining the like terms:
[tex]\[
(-3 + 2i) - 48 - 8i
\][/tex]
[tex]\[
-3 - 48 + 2i - 8i
\][/tex]
[tex]\[
-51 - 6i
\][/tex]
So, the simplified expression is:
[tex]\[
-51 - 6i
\][/tex]
This matches our option, so the correct answers are:
[tex]\[
\boxed{-51-6i}
\][/tex]
Now, checking other equivalent expressions:
1. [tex]\[
-51 - 6i \quad (\text{already determined as correct})
\][/tex]
2. [tex]\[
-3-2i-2(24)+8i \quad = \quad -3 - 2i - 48 + 8i \quad = \quad -51 + 6i \quad (\text{not correct})
\][/tex]
3. [tex]\[
45 + 10i \quad (\text{not equivalent})
\][/tex]
4. [tex]\[
-3 + 2i + 2(24) + 8i \quad = \quad -3 + 2i + 48 + 8i \quad = \quad 45 + 10i \quad (\text{not correct})
\][/tex]
5. [tex]\[
-3 + 2i - 2(24) - 8i \quad = \quad -3 + 2i - 48 - 8i \quad = \quad -51 - 6i \quad (\text{correct})
\][/tex]
6. [tex]\[
-51 + 6i \quad (\text{not correct})
\][/tex]
Thus, the correct selections are:
[tex]\[
\boxed{-51-6i \quad \text{and} \quad -3+2i-2(24)-8i}
\][/tex]
