1. **State the problem:** We need to find the value of $x$ such that the sum of the number $405$ in base $x$ and the number $43$ in base $8$ equals $184$ in base $10$. 2. **Write the equation:** $$405_x + 43_8 = 184_{10}$$ 3. **Convert known bases to base 10:** - Convert $43_8$ to base 10: $$4 \times 8^1 + 3 \times 8^0 = 4 \times 8 + 3 = 32 + 3 = 35$$ - $184_{10}$ is already in base 10. 4. **Express $405_x$ in base 10:** $$4 \times x^2 + 0 \times x + 5 = 4x^2 + 5$$ 5. **Set up the equation in base 10:** $$4x^2 + 5 + 35 = 184$$ 6. **Simplify the equation:** $$4x^2 + 40 = 184$$ $$4x^2 = 184 - 40$$ $$4x^2 = 144$$ 7. **Solve for $x^2$:** $$x^2 = \frac{144}{4} = 36$$ 8. **Find $x$:** $$x = \sqrt{36} = 6$$ 9. **Check the base validity:** Since the digits in $405_x$ are 4, 0, and 5, the base $x$ must be greater than 5. Our solution $x=6$ satisfies this. **Final answer:** $$\boxed{6}$$