The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number?
Let $z$ be the image of $2i$ under the dilation.

[asy]
unitsize(0.5 cm);

pair C, P, Q;

C = (-1,4);
P = (0,2);
Q = (-3,8);
draw((-5,0)--(5,0));
draw((0,-1)--(0,10));
draw(P--Q,dashed);

dot("$-1 + 4i$", C, SW);
dot("$2i$", P, E);
dot("$-3 + 8i$", Q, NW);
[/asy]

Since the dilation is centered at $-1 + 4i,$ with scale factor $-2,$
\[z - (-1 + 4i) = (-2)(2i - (-1 + 4i)).\]Solving, we find $z = \boxed{-3 + 8i}.$
-3+8i
Level 3
Precalculus
Problem: The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the image of the point $2i$ under the dilation centered at $-1 + 4i$ with a scale factor of $-2$, we first need to understand the process of dilation in the complex plane. 

Dilation involves two main steps: translating the point to the origin, applying the scale factor, and then translating it back. 

1. **Translation to the Origin**: We subtract the center of dilation from the point we are dilating. The center is $-1 + 4i$ and the point is $2i$. 
   - Calculation: 
   - $2i - (-1 + 4i) = 2i + 1 - 4i = 1 - 2i$.

2. **Applying the Scale Factor**: The scale factor is $-2$. We multiply the translated point by this scale factor.
   - Calculation:
   - $-2 \cdot (1 - 2i) = -2 \cdot 1 + (-2) \cdot (-2i) = -2 + 4i$.

3. **Translation Back**: Now we add the center of dilation back to the scaled point.
   - Calculation:
   - $(-2 + 4i) + (-1 + 4i) = -2 - 1 + 4i + 4i = -3 + 8i$.

Thus, the image of the point $2i$ under the dilation is $-3 + 8i$.<|END_OF_LATENT|> The final answer is $-3+8i$.