1. **State the problem:** Find the coordinates of the image (reflection) of the point $(-4,5)$ about the line $x + 3y - 1 = 0$. 2. **Formula and explanation:** The reflection of a point $(x_0,y_0)$ about a line $Ax + By + C = 0$ can be found using the formula: $$x' = x_0 - \frac{2A(Ax_0 + By_0 + C)}{A^2 + B^2}, \quad y' = y_0 - \frac{2B(Ax_0 + By_0 + C)}{A^2 + B^2}$$ where $(x', y')$ are the coordinates of the reflected point. 3. **Identify values:** Here, $A=1$, $B=3$, $C=-1$, and the point is $(x_0,y_0) = (-4,5)$. 4. **Calculate the numerator:** $$Ax_0 + By_0 + C = 1 \times (-4) + 3 \times 5 - 1 = -4 + 15 - 1 = 10$$ 5. **Calculate the denominator:** $$A^2 + B^2 = 1^2 + 3^2 = 1 + 9 = 10$$ 6. **Calculate $x'$:** $$x' = -4 - \frac{2 \times 1 \times 10}{10} = -4 - 2 = -6$$ 7. **Calculate $y'$:** $$y' = 5 - \frac{2 \times 3 \times 10}{10} = 5 - 6 = -1$$ 8. **Final answer:** The coordinates of the image of the point $(-4,5)$ about the line $x + 3y - 1 = 0$ are $$\boxed{(-6, -1)}$$.