Using the angle bisector theorem solve for x. show all work.

X = 6.

The angle bisector theorem tells us that the ratio of the lengths of the line segments on the side the bisector intersects is the same as the ratio of the lengths of the sides adjacent to the intersected side. So using the theorem and the diagram, we can create this equation, and then solve for x.
(4x+1)/15 = 5/3
15*(4x+1)/15 = 15*5/3
4x+1 = 5*5
4x+1 = 25
4x+1-1 = 25-1
4x = 24
4x/4 = 24/4
x = 6