How many inches would a point on the outer edge of the large gear travel in a 150º rotation? if the larger gear is 4 inches. explain the formula .

Are there two gears here?  You imply that by using the wording "larger gear."

I will assume that you have only one gear, and that its radius is 4 inches.  How many inches does a point on the outer edge of the gear move if the central angle increases by 150 degrees?

Use the arc length formula:  s = r * theta, where theta is the central angle in radians, not degrees).

You must convert 150 degrees to radians.  Remember how to do that?  

150 degrees = 5 pi/6 rad (please check and state whether this is correct or not).

Then the arc length in question is s = r (theta), or s = (4 inches) (5 pi/6 rad)

This comes out to s = (20/6) inches, or s = 10/3 inches, or 3 2/3 inches.