<Question>
A $90^\circ$ rotation around $-4 - 5i$ in the clockwise direction is applied to $3$ (as a complex number).  What is the resulting complex number?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out the result of rotating the complex number 3 (which is 3 + 0i) 90 degrees clockwise around the point -4 - 5i. Hmm, complex rotations... Let me recall how that works. 

First, I remember that rotations in the complex plane can be done by multiplying by a complex exponential. A 90-degree rotation is a common one, so maybe I can use that. But since the rotation isn't around the origin, but around the point -4 -5i, I need to translate the complex plane so that the center of rotation becomes the origin, perform the rotation, and then translate back.

Let me break it down step by step.

<prompt_1.1> Plan the steps to translate the complex plane so that the center of rotation becomes the origin, perform the rotation, and translate back. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, translate the original point (3) so that the center of rotation (-4 -5i) becomes the origin. To do this, I subtract the center of rotation from the original point. So, the translated point would be 3 - (-4 -5i). Let me compute that: 3 +4 +5i = 7 +5i. So, the translated point is 7 +5i.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to rotate this translated point 90 degrees clockwise around the origin. I remember that a 90-degree rotation clockwise is equivalent to multiplying by -i. Wait, let me confirm that.
A 90-degree counterclockwise rotation is multiplying by i, so clockwise would be the negative of that, which is -i. So, if I have a complex number z, then a 90-degree clockwise rotation is -i*z. Let me check with a simple example.

<prompt_2.1> Verify that a 90-degree clockwise rotation is equivalent to multiplying by -i using specific examples. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
If z is 1 + 0i, rotating 90 degrees clockwise should take it to 0 + (-i). So, multiplying 1 by -i gives -i, which is 0 -1i. That's correct. So yes, multiplying by -i works for 90 degrees clockwise.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
So, applying that to our translated point 7 +5i. Let's multiply (7 +5i) by -i. Let me compute that:

<prompt_3.1> Calculate the result of rotating the translated point 7 + 5i by 90 degrees clockwise using complex multiplication. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
(7 +5i)*(-i) = 7*(-i) +5i*(-i) = -7i -5i². Since i² is -1, this becomes -7i -5*(-1) = -7i +5. So, that's 5 -7i.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Now, we need to translate back by adding the center of rotation. The center was -4 -5i, so we add that to the rotated point.
So, 5 -7i + (-4 -5i) = (5 -4) + (-7i -5i) = 1 -12i.

Wait, is that right? Let me check each step again to make sure.

<prompt_4.1> Verify each step of the calculation to ensure the result is correct. </prompt_4.1>

But let me think again—if I have a point and I translate it by subtracting the center, rotate, then add the center back, that should give the correct result. Let me visualize this.

<prompt_4.2> Visualize the transformation process to confirm the correctness of the result. </prompt_4.2>

Alternatively, maybe I can think of another way. Let me consider the general formula for rotation around a point. If you want to rotate a point z around a point c by angle θ, the formula is: 

rotated point = (z - c) * e^{iθ} + c

In this case, θ is -90 degrees because it's clockwise. So θ = -π/2 radians. The exponential e^{-iπ/2} is cos(-π/2) + i sin(-π/2) = cos(π/2) - i sin(π/2) = 0 - i*1 = -i. So that's consistent with what I did earlier. So, the formula becomes:

rotated point = (z - c)*(-i) + c

So, substituting z = 3, c = -4 -5i:
rotated point = (3 - (-4 -5i))*(-i) + (-4 -5i)
= (3 +4 +5i)*(-i) + (-4 -5i)
= (7 +5i)*(-i) + (-4 -5i)
= (-7i -5i²) + (-4 -5i)
= (-7i +5) + (-4 -5i)
= 5 -4 + (-7i -5i)
= 1 -12i

So that's the same result as before.

Alternatively, to check again, let's use coordinates. Let's represent the complex number as coordinates in the plane. The original point 3 is (3, 0). The center of rotation is (-4, -5). To rotate (3,0) 90 degrees clockwise around (-4, -5), we can translate the point to origin-centered coordinates, rotate, then translate back.

<prompt_4.3> Verify the result using a coordinate geometry approach to ensure consistency. </prompt_4.3>

I think that's thorough. Let me just check once more.

<prompt_4.4> Perform a final verification of the steps and conclude the solution. </prompt_4.4>
</Plan_4>

<Execution_4>
<execution_4.1>
First step: translating 3 by subtracting -4 -5i. So 3 - (-4 -5i) = 3 +4 +5i =7 +5i. Correct. Then multiplying by -i: (7 +5i)(-i) = -7i -5i². Since i² is -1, that's -7i +5. So 5 -7i. Then translating back by adding -4 -5i: 5 -4 =1, and -7i -5i = -12i. So 1 -12i.
</execution_4.1>

<execution_4.2>
Original point: 3 (which is (3,0) in the plane). Center of rotation: -4 -5i (which is (-4, -5)). Translating the original point by subtracting the center: (3 - (-4), 0 - (-5)) = (7,5). So that's the point (7,5) relative to the origin. Then, rotating 90 degrees clockwise would transform (x, y) to (y, -x). Wait, is that right? Wait, when rotating 90 degrees clockwise, the transformation is (x, y) → (y, -x). Let me confirm. Original point (x, y), after rotation, the new coordinates (x', y') are such that the rotation matrix R is:

For 90 degrees clockwise, the matrix is [ [0, 1], [ -1, 0] ]. So, applying 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, translating back by adding the center (-4, -5):

(5 + (-4), -7 + (-5)) = (1, -12). So the resulting point is (1, -12), which corresponds to the complex number 1 -12i.
</execution_4.2>

<execution_4.3>
Translating (3,0) by subtracting (-4, -5) gives (3 - (-4), 0 - (-5)) = (7,5). 

A 90-degree clockwise rotation of a point (x,y) around the origin becomes (y, -x). Wait, let me confirm. If you have a point (x,y) and rotate it 90 degrees clockwise, the new coordinates become (y, -x). Let me see: original point (x,y) becomes (y, -x). So, (7,5) becomes (5, -7). 

Then, translating back by adding (-4, -5): (5 + (-4), -7 + (-5)) = (1, -12). Which corresponds to the complex number 1 -12i.
</execution_4.3>

<execution_4.4>
Original point: 3 (as a complex number). Center: -4 -5i. Translate: 3 - (-4 -5i) = 3 +4 +5i = 7 +5i. Rotate 90 degrees clockwise: multiply by -i. (7 +5i)(-i) = -7i -5i² = -7i +5. Translate back: add (-4 -5i). So 5 -7i -4 -5i = (5-4) + (-7i -5i) = 1 -12i. 

Yes, that seems consistent. So the final result is 1 -12i.
</execution_4.4>
</Execution_4>

<Final_answer>
Therefore, the resulting complex number after a $90^\circ$ clockwise rotation around $-4 - 5i$ applied to $3$ is $\boxed{1 - 12i}$.
</Final_answer>
</think>