1. **State the problem:** Simplify the complex fraction $$\frac{3+2i}{4-i}$$ by dividing. 2. **Formula and rule:** To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $$4 - i$$ is $$4 + i$$. 3. **Multiply numerator and denominator:** $$\frac{3+2i}{4-i} \times \frac{4+i}{4+i} = \frac{(3+2i)(4+i)}{(4-i)(4+i)}$$ 4. **Expand numerator:** $$(3)(4) + (3)(i) + (2i)(4) + (2i)(i) = 12 + 3i + 8i + 2i^2$$ Since $$i^2 = -1$$, substitute: $$12 + 3i + 8i + 2(-1) = 12 + 11i - 2 = 10 + 11i$$ 5. **Expand denominator:** $$(4)(4) + (4)(i) - (i)(4) - (i)(i) = 16 + 4i - 4i - i^2$$ Simplify: $$16 + 0 - (-1) = 16 + 1 = 17$$ 6. **Write the fraction:** $$\frac{10 + 11i}{17} = \frac{10}{17} + \frac{11}{17}i$$ 7. **Final answer:** $$\boxed{\frac{10}{17} + \frac{11}{17}i}$$