1. **State the problem:** Solve the exponential equation $25^{2x-1} = 125^{3x+4}$ for $x$. 2. **Recall the formula and rules:** Both 25 and 125 can be expressed as powers of 5 since $25 = 5^2$ and $125 = 5^3$. 3. **Rewrite the equation using base 5:** $$25^{2x-1} = (5^2)^{2x-1} = 5^{2(2x-1)} = 5^{4x-2}$$ $$125^{3x+4} = (5^3)^{3x+4} = 5^{3(3x+4)} = 5^{9x+12}$$ 4. **Set the exponents equal since bases are the same:** $$4x - 2 = 9x + 12$$ 5. **Solve for $x$:** $$4x - 2 = 9x + 12$$ $$4x - 9x = 12 + 2$$ $$-5x = 14$$ $$x = -\frac{14}{5}$$ 6. **Final answer:** $$x = -\frac{14}{5}$$