The problem asks for the coordinates of $B'$ after a dilation of triangle $ABC$ with a scale factor of 3. The original coordinates of $B$ are $B(3, -2)$.

A dilation with a scale factor $k$ multiplies both the $x$- and $y$-coordinates of a point by $k$. Therefore, for point $B(3, -2)$ with a scale factor of 3, the coordinates of $B'$ will be:

$B'(x', y') = (3 \times 3, 3 \times -2) = (9, -6)$

Thus, the coordinates of $B'$ are $(9, -6)$.

The correct answer is $(9, -6)$, which matches the third option.

Would you like further details on dilation or any other part of this concept?

Here are five questions that extend this concept:

1. What are the coordinates of $A'$ after a dilation with the same scale factor?
2. How would the coordinates of $C'$ change with a scale factor of 2 instead of 3?
3. What happens to the shape and size of a triangle after a dilation?
4. How would you reverse a dilation? What is the scale factor needed for that?
5. What is the effect of a negative scale factor on dilation?

Tip: To perform dilations, always multiply both coordinates of a point by the scale factor.