question

Use Pythagoras to get the length of the segment as

$$\sqrt{(x--1)^2+(0-4)^2}=\sqrt{(x+1)^2+16}$$  So we must have 

$$\sqrt{(x+1)^2+16}=5$$ or 

$$(x+1)^2+16=25\\\\ (x+1)^2=9\\\\ x+1=\pm3\\\\ x=\pm3-1$$ so x = 2 and x = -4.