1. **State the problem:** We are given two intersecting lines forming vertical angles. One angle measures 104° and the opposite angle measures $4(x+1)^\circ$. We need to find the value of $x$. 2. **Recall the angle relationship:** Vertical angles are equal when two lines intersect. So, $$104^\circ = 4(x+1)^\circ$$ 3. **Set up the equation:** $$104 = 4(x+1)$$ 4. **Solve for $x$:** Divide both sides by 4: $$\frac{104}{\cancel{4}} = \frac{4(x+1)}{\cancel{4}}$$ $$26 = x + 1$$ 5. **Isolate $x$:** $$x = 26 - 1$$ $$x = 25$$ **Final answer:** $$\boxed{25}$$