Given points are
$A(4,3,2), B(5,4,6), \mathrm{C}(-1,-1,5)$
$A B=\sqrt{(5-4)^2+(4-3)^2+(6-2)^2}$
$=\sqrt{(1)^2+(1)^2+(4)^2}=\sqrt{18}$
$A B=3 \sqrt{2}$
$A C=\sqrt{(-1-4)^2+(-1-3)^2+(5-2)^2}$
$=\sqrt{(-5)^2+(-4)^2+(3)^2}=\sqrt{25+16+9}=\sqrt{50}$
$A C=5 \sqrt{2}$
$A B: A C=3 \sqrt{2}: 5 \sqrt{2}=3: 5$
Let say the bisector of angle $A$ meet the side $B C$ that point $D\left(x_D, y_D, z_D\right)$ and $D$ divides $B C$ side in ratio 3: 5 internally
$x_D=\frac{m x_C+n x_B}{m+n}=\frac{3 \times(-1)+5 \times 5}{3+5}=\frac{22}{8}$
$y_D=\frac{m y_C+n y_B}{m+n}=\frac{3 \times(-1)+5 \times(4)}{3+5}=\frac{17}{8}$
$z_D=\frac{m z_C+n z_B}{m+n}=\frac{3 \times(5)+5 \times(6)}{3+5}=\frac{45}{8}$
Therefore, coordinates of point $D$ is $\left(\frac{22}{8}, \frac{17}{8}, \frac{45}{8}\right)$.