1. **State the problem:** Solve for the complex number $z$ in the equation $$(3-5i)(z+2i) = i.$$ 2. **Use the distributive property:** Expand the left side: $$ (3-5i)(z+2i) = (3-5i)z + (3-5i)(2i). $$ 3. **Calculate the product $(3-5i)(2i)$:** $$ (3-5i)(2i) = 3 \cdot 2i - 5i \cdot 2i = 6i - 10i^2. $$ Since $i^2 = -1$, this becomes $$ 6i - 10(-1) = 6i + 10 = 10 + 6i. $$ 4. **Rewrite the equation:** $$ (3-5i)z + 10 + 6i = i. $$ 5. **Isolate $(3-5i)z$:** $$ (3-5i)z = i - 10 - 6i = -10 - 5i. $$ 6. **Solve for $z$ by dividing both sides by $(3-5i)$:** $$ z = \frac{-10 - 5i}{3 - 5i}. $$ 7. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate of the denominator $(3 + 5i)$: $$ z = \frac{(-10 - 5i)(3 + 5i)}{(3 - 5i)(3 + 5i)}. $$ 8. **Calculate the denominator:** $$ (3 - 5i)(3 + 5i) = 3^2 - (5i)^2 = 9 - (-25) = 9 + 25 = 34. $$ 9. **Calculate the numerator:** $$ (-10)(3) + (-10)(5i) + (-5i)(3) + (-5i)(5i) = -30 - 50i - 15i - 25i^2. $$ Since $i^2 = -1$, $$ -25i^2 = -25(-1) = 25. $$ So numerator becomes $$ -30 - 65i + 25 = (-30 + 25) - 65i = -5 - 65i. $$ 10. **Write $z$ as:** $$ z = \frac{-5 - 65i}{34} = -\frac{5}{34} - \frac{65}{34}i. $$ 11. **Simplify fractions:** $$ -\frac{5}{34} = -0.1470588..., \quad -\frac{65}{34} = -1.9117647... $$ So approximately $$ z \approx -0.147 - 1.912i. $$ **Final answer:** $$ z = -\frac{5}{34} - \frac{65}{34}i. $$