frrom (1,4) you travel 5root2 units by making 135degree with positive x-axis(anticlockwise) and then 4 units by making 120degree with positive x-axis clockwise to reach Q.find coordinate of point Q

let the coordinates of point A be (1,4). 
let travelling along 135 deg with the positive direction of x-axis at distance 5 root 2, we reach to the point $B_1 or B_2$
 then the coordinates of the points are given by
$$\begin{gathered} \frac{x-1}{\cos 135}=\frac{y-4}{\sin 135}=\pm 5\sqrt{2} \\ \frac{x-1}{-1/\sqrt{2}}=\frac{y-4}{1/\sqrt{2}}=\pm 5\sqrt{2} \\ x=1-\frac{1}{\sqrt{2}}.(\pm 5\sqrt{2}) and y=4+\frac{1}{\sqrt{2}}.(\pm 5\sqrt{2}) \\ x=1-(\pm 5) and y=4\pm 5 \\ x=1-5 and y=4+5 or x=1+5 and y=4-5 \\ x=-4 and y=9 or x=6 and y=-1 \\ then B_1(-4,9) and B_2(6,-1) \end{gathered}$$
now from the point B1, if we move 4 units making 120 deg with positive x-axis clockwise.
then the angle made in anticlockwise direction is 180-120=60 deg
corresponding to point B1, we get two points Q1 and Q2,
the coordinates of the points Q1 and Q2 are given by:
$$\begin{gathered} \frac{x+4}{\cos 60}=\frac{y-9}{\sin 60}=\pm 4 \\ \frac{x+4}{1/2}=\frac{y-9}{\sqrt{3}/2}=\pm 4 \\ x=-4+\frac{1}{2}.(\pm 4) and y=9+\frac{\sqrt{3}}{2}.(\pm 4) \\ x=-4\pm 2 and y=9\pm 2\sqrt{3} \\ x=-4+2 and y=9+2\sqrt{3} or x=-4-2 and y=9-2\sqrt{3} \\ x=-2 and y=9+2\sqrt{3} or x=-6 and y=9-2\sqrt{3} \\ {Q}_1 (-2,9+2\sqrt{3}) and Q_2(-6 ,9-2\sqrt{3}) \end{gathered}$$
corresponding to point B2, we wil get two more points:
the coordinates are given by:
$$\begin{gathered} \frac{x-6}{\cos 60}=\frac{y+1}{\sin 60}=\pm 4 \\ x=6\pm 4*\frac{1}{2} and y=-1\pm 4*\frac{\sqrt{3}}{2} \\ x=6\pm 2 and y=-1\pm 2\sqrt{3} \\ x=6+2 and y=-1+2\sqrt{3} or x=6-2 and y=-1-2\sqrt{3} \\ x=8 and y=-1+2\sqrt{3} or x=4 and y=-1-2\sqrt{3} \\ {Q}_3 (8, -1+2\sqrt{3}) and Q_4 (4,-1-2\sqrt{3}) \end{gathered}$$

hope this helps you
cvb