1. **State the problem:** We need to find the value of $x$ given that the exterior angle of a regular octagon is $(4x - 3)^\circ$. 2. **Recall the formula for exterior angles of regular polygons:** The measure of each exterior angle of a regular polygon with $n$ sides is given by $$\text{Exterior angle} = \frac{360^\circ}{n}$$ 3. **Apply the formula for an octagon:** Since an octagon has $n=8$ sides, $$\text{Exterior angle} = \frac{360^\circ}{8} = 45^\circ$$ 4. **Set up the equation:** Given the exterior angle is $(4x - 3)^\circ$, we have $$4x - 3 = 45$$ 5. **Solve for $x$:** $$4x - 3 = 45$$ $$4x = 45 + 3$$ $$4x = 48$$ $$x = \frac{48}{4}$$ $$x = 12$$ 6. **Use cancellation notation for division:** $$x = \frac{\cancel{48}}{\cancel{4}} = 12$$ **Final answer:** $x = 12$