1. **State the problem:** We are given that five times the number $34_x$ equals the number $214_x$, and we need to find the base $x$. 2. **Understand the notation:** The subscript $x$ means the number is in base $x$. For example, $34_x$ means $3 \times x + 4$ in decimal. 3. **Write the equation:** $$5 \times (3x + 4) = 2x^2 + 1x + 4$$ 4. **Expand and simplify:** $$15x + 20 = 2x^2 + x + 4$$ 5. **Bring all terms to one side:** $$0 = 2x^2 + x + 4 - 15x - 20$$ $$0 = 2x^2 - 14x - 16$$ 6. **Divide the entire equation by 2 to simplify:** $$x^2 - 7x - 8 = 0$$ 7. **Solve the quadratic equation using the quadratic formula:** $$x = \frac{7 \pm \sqrt{(-7)^2 - 4 \times 1 \times (-8)}}{2} = \frac{7 \pm \sqrt{49 + 32}}{2} = \frac{7 \pm \sqrt{81}}{2}$$ 8. **Calculate the roots:** $$x = \frac{7 + 9}{2} = 8 \quad \text{or} \quad x = \frac{7 - 9}{2} = -1$$ 9. **Interpret the results:** Base $x$ must be a positive integer greater than the largest digit in the number. The digits are 3 and 4, so $x > 4$. The valid solution is $x = 8$. **Final answer:** $$\boxed{8}$$