1. **Problem statement:** Given complex numbers $Z_1 = 2 - 3i$, $Z_2 = 6 + i$, $Z_3 = 2 + 4i$, and $Z_4 = 5 - i$, find the quotient $\frac{Z_2}{Z_1}$. 2. **Formula and rules:** To divide two complex numbers, multiply numerator and denominator by the conjugate of the denominator: $$\frac{a+bi}{c+di} = \frac{(a+bi)(c - di)}{(c+di)(c - di)} = \frac{(a+bi)(c - di)}{c^2 + d^2}$$ 3. **Apply to $\frac{Z_2}{Z_1}$:** $$\frac{6 + i}{2 - 3i} = \frac{(6 + i)(2 + 3i)}{(2 - 3i)(2 + 3i)}$$ 4. **Calculate denominator:** $$(2 - 3i)(2 + 3i) = 2^2 + 3^2 = 4 + 9 = 13$$ 5. **Calculate numerator:** $$(6 + i)(2 + 3i) = 6 \times 2 + 6 \times 3i + i \times 2 + i \times 3i = 12 + 18i + 2i + 3i^2$$ 6. **Simplify numerator:** Since $i^2 = -1$, $$12 + 18i + 2i + 3(-1) = 12 + 20i - 3 = 9 + 20i$$ 7. **Write final expression:** $$\frac{9 + 20i}{13} = \frac{9}{13} + \frac{20}{13}i$$ 8. **Answer:** $$\boxed{\frac{Z_2}{Z_1} = \frac{9}{13} + \frac{20}{13}i}$$ This completes the first problem.