Okay, so I need to figure out what happens when I rotate the complex number 3 (which is 3 + 0i) 90 degrees clockwise around the point -4 - 5i. Hmm, complex rotations can be a bit tricky, but let me recall how rotations work in the complex plane.

First, I remember that a rotation in the complex plane can be achieved by multiplying by a complex number of unit magnitude. Specifically, a rotation by θ radians clockwise around the origin is equivalent to multiplying by e^{-iθ}. Since we're dealing with a 90-degree rotation, that's π/2 radians. So the rotation factor would be e^{-iπ/2} which is cos(π/2) - i sin(π/2) = 0 - i*1 = -i. So multiplying by -i should give a 90-degree clockwise rotation around the origin.

But wait, this rotation is around the point -4 -5i, not the origin. So I need to translate the complex plane so that the center of rotation becomes the origin, perform the rotation, and then translate back. That makes sense. Let me outline the steps:

1. Translate the original complex number 3 by subtracting the center of rotation, which is -4 -5i. So this would be 3 - (-4 -5i) = 3 +4 +5i = 7 +5i.

2. Rotate this translated complex number 90 degrees clockwise. As we determined, that's multiplying by -i. So (7 +5i) * (-i).

3. Translate back by adding the center of rotation again. So after rotation, add (-4 -5i) to the result.

Let me do each step carefully.

First, translation: 3 - (-4 -5i) = 3 +4 +5i = 7 +5i. That seems right.

Next, rotation: (7 +5i)*(-i). Let's compute that. Multiply each term:

7*(-i) = -7i

5i*(-i) = -5i². But i² is -1, so this becomes -5*(-1) = 5.

So combining the terms: -7i +5 = 5 -7i.

Then, translating back: 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i.

Wait, so the result is 1 -12i? Let me check each step again.

First translation: 3 - (-4 -5i) = 3 +4 +5i = 7 +5i. Correct.

Multiply by -i: (7 +5i)*(-i). Let's distribute:

7*(-i) = -7i

5i*(-i) = -5i² = -5*(-1) = 5. Then adding those together: 5 -7i. Then add back the translation: 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i. Hmm, that seems right.

But let me verify using another method to ensure I didn't make a mistake. Another way to think about rotation around a point is to use the formula:

If you have a point z and you want to rotate it around a point c by angle θ, the formula is:

z' = (z - c) * e^{-iθ} + c

Which is exactly what I did. So θ is 90 degrees, which is π/2. So e^{-iπ/2} = cos(-π/2) + i sin(-π/2) = 0 - i, which is -i. So yes, that's correct.

So plugging in:

z = 3, c = -4 -5i

z' = (3 - (-4 -5i)) * (-i) + (-4 -5i)

= (7 +5i)*(-i) + (-4 -5i)

= (-7i -5i²) + (-4 -5i)

= (-7i -5*(-1)) + (-4 -5i)

= (-7i +5) + (-4 -5i)

= 5 -4 + (-7i -5i)

= 1 -12i

Yes, same result. So that seems correct.

Alternatively, maybe I can use coordinates. Let's represent the complex numbers as points in the plane.

Original point: 3 is (3, 0).

Center of rotation: -4 -5i is (-4, -5).

To rotate (3,0) 90 degrees clockwise around (-4, -5):

First, translate the point so that the center becomes the origin. Subtract (-4, -5) from (3,0):

(3 - (-4), 0 - (-5)) = (7,5).

Now, rotate (7,5) 90 degrees clockwise around the origin. A 90-degree clockwise rotation of a point (x,y) is (y, -x). Wait, is that right?

Wait, let me recall. A 90-degree rotation can be represented as a matrix. For clockwise rotation, the matrix is:

[cosθ  sinθ]
[-sinθ cosθ]

But for θ = 90 degrees, cosθ = 0, sinθ = 1.

So the rotation matrix becomes:

[0  1]
[-1 0]

So applying this to (x,y):

x' = 0*x + 1*y = y

y' = -1*x + 0*y = -x

So yes, (x,y) becomes (y, -x). So (7,5) becomes (5, -7).

Then, translate back by adding (-4, -5):

(5 + (-4), -7 + (-5)) = (1, -12). So the point is (1, -12), which corresponds to the complex number 1 -12i. That's the same result as before. So that confirms it.

Therefore, the answer should be 1 -12i. But let me just check once more with another method to be absolutely sure.

Another way: The formula for rotation around a point in complex plane is:

w = (z - c) * e^{-iθ} + c

Where θ is 90 degrees. So in this case, θ = π/2, so e^{-iπ/2} = -i as before.

So (3 - (-4 -5i)) = 7 +5i

Multiply by -i: 7*(-i) +5i*(-i) = -7i -5i² = -7i +5 (since i² = -1)

Then add the center: 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i. Yep, same answer.

Alternatively, using vectors. The vector from the center to the point is 3 - (-4 -5i) = 7 +5i. This vector is (7,5). Rotating it 90 degrees clockwise gives (5, -7). Adding back the center's coordinates (-4, -5) gives (-4 +5, -5 + (-7)) = (1, -12). So (1, -12) is the final point, which is 1 -12i.

All methods converge to the same result. Therefore, I'm confident the answer is 1 -12i.

**Final Answer**
The resulting complex number is \boxed{1 - 12i}.
Therefore, the resulting complex number after the rotation is $\boxed{1 - 12i}$.
</think>