1. **State the problem:** Solve for the complex number $z$ in the equation $$(3-5i)(z+2) = i.$$\n\n2. **Use the distributive property:** Expand the left side using the distributive property: $$(3-5i)(z+2) = (3-5i)z + (3-5i)2.$$\n\n3. **Rewrite the equation:** $$ (3-5i)z + 2(3-5i) = i.$$\n\n4. **Calculate the constant term:** $$2(3-5i) = 6 - 10i.$$\n\n5. **Rewrite the equation:** $$ (3-5i)z + 6 - 10i = i.$$\n\n6. **Isolate the term with $z$:** $$ (3-5i)z = i - 6 + 10i = -6 + 11i.$$\n\n7. **Solve for $z$ by dividing both sides:** $$ z = \frac{-6 + 11i}{3 - 5i}.$$\n\n8. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate of the denominator $3 + 5i$: $$ z = \frac{(-6 + 11i)(3 + 5i)}{(3 - 5i)(3 + 5i)}.$$\n\n9. **Calculate the denominator:** $$(3 - 5i)(3 + 5i) = 3^2 - (5i)^2 = 9 - (-25) = 9 + 25 = 34.$$\n\n10. **Calculate the numerator:** $$(-6)(3) + (-6)(5i) + (11i)(3) + (11i)(5i) = -18 - 30i + 33i + 55i^2.$$\n\n11. **Simplify numerator terms:** $$-18 + 3i + 55(-1) = -18 + 3i - 55 = -73 + 3i.$$\n\n12. **Write the fraction:** $$ z = \frac{-73 + 3i}{34} = \frac{-73}{34} + \frac{3}{34}i.$$\n\n**Final answer:** $$\boxed{z = -\frac{73}{34} + \frac{3}{34}i}.$$