1. **State the problem:** Solve the equation $5(3 - 2i) + 2i(4 + 6i) = 0$ for the imaginary unit $i$. 2. **Recall the property of $i$:** $i^2 = -1$. 3. **Expand the terms:** $$5(3 - 2i) = 15 - 10i$$ $$2i(4 + 6i) = 8i + 12i^2$$ 4. **Combine the expanded terms:** $$15 - 10i + 8i + 12i^2 = 0$$ 5. **Simplify the imaginary terms:** $$15 - 2i + 12i^2 = 0$$ 6. **Substitute $i^2 = -1$:** $$15 - 2i + 12(-1) = 0$$ $$15 - 2i - 12 = 0$$ 7. **Simplify the constants:** $$3 - 2i = 0$$ 8. **Separate real and imaginary parts:** Real part: $3 = 0$ (which is false) Imaginary part: $-2i = 0$ implies $i = 0$ (which contradicts the definition of $i$) **Conclusion:** The equation simplifies to a contradiction, so there is no solution for $i$ that satisfies this equation. The original expression is an identity involving $i$, not an equation to solve for $i$. Final simplified expression: $$3 - 2i = 0$$