1. **Problem statement:** Find the coordinates of point P that divides the line segment AB in the ratio 2:3, where A has coordinates $(3,-1,-4)$ and B has coordinates $(8,-6,6)$. 2. **Formula used:** If a point P divides the segment AB in the ratio $m:n$, then the coordinates of P are given by: $$P = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right)$$ where $A = (x_1, y_1, z_1)$ and $B = (x_2, y_2, z_2)$. 3. **Apply the formula:** Here, $m=2$, $n=3$, $A=(3,-1,-4)$, and $B=(8,-6,6)$. Calculate each coordinate: $$x_P = \frac{2 \times 8 + 3 \times 3}{2+3} = \frac{16 + 9}{5} = \frac{25}{5} = 5$$ $$y_P = \frac{2 \times (-6) + 3 \times (-1)}{5} = \frac{-12 - 3}{5} = \frac{-15}{5} = -3$$ $$z_P = \frac{2 \times 6 + 3 \times (-4)}{5} = \frac{12 - 12}{5} = \frac{0}{5} = 0$$ 4. **Final answer:** The coordinates of point P are $(5, -3, 0)$.