1. **Problem statement:** Find the coordinates of point C that divides the line segment AB externally in the ratio $|AC| : |CB| = 4 : 1$, where $A=(-2,-1)$ and $B=(3,4)$. 2. **Formula for external division:** If a point $C$ divides the segment $AB$ externally in the ratio $m:n$, then $$C = \left( \frac{m x_B - n x_A}{m - n}, \frac{m y_B - n y_A}{m - n} \right)$$ where $A=(x_A,y_A)$ and $B=(x_B,y_B)$. 3. **Apply the formula:** Here, $m=4$, $n=1$, $x_A=-2$, $y_A=-1$, $x_B=3$, $y_B=4$. Calculate the $x$-coordinate: $$x_C = \frac{4 \times 3 - 1 \times (-2)}{4 - 1} = \frac{12 + 2}{3} = \frac{14}{3} \approx 4.67$$ Calculate the $y$-coordinate: $$y_C = \frac{4 \times 4 - 1 \times (-1)}{4 - 1} = \frac{16 + 1}{3} = \frac{17}{3} \approx 5.67$$ 4. **Interpretation:** Point $C$ lies outside the segment $AB$ because the ratio is external division. 5. **Final answer:** The coordinates of $C$ are $$\boxed{\left( \frac{14}{3}, \frac{17}{3} \right)}$$ or approximately $(4.67, 5.67)$.