1. **Convert $ \frac{3 - 5i}{2 + 7i} $ into the form $ a + bi $.** 2. To convert a complex fraction into the form $ a + bi $, multiply numerator and denominator by the conjugate of the denominator. 3. The conjugate of $ 2 + 7i $ is $ 2 - 7i $. 4. Multiply numerator and denominator: $$\frac{3 - 5i}{2 + 7i} \times \frac{2 - 7i}{2 - 7i} = \frac{(3 - 5i)(2 - 7i)}{(2 + 7i)(2 - 7i)}$$ 5. Expand numerator: $$3 \times 2 = 6$$ $$3 \times (-7i) = -21i$$ $$-5i \times 2 = -10i$$ $$-5i \times (-7i) = +35i^2$$ 6. Sum numerator terms: $$6 - 21i - 10i + 35i^2 = 6 - 31i + 35(-1) = 6 - 31i - 35 = -29 - 31i$$ 7. Denominator: $$ (2)^2 - (7i)^2 = 4 - 49i^2 = 4 - 49(-1) = 4 + 49 = 53$$ 8. So the expression is: $$\frac{-29 - 31i}{53} = -\frac{29}{53} - \frac{31}{53}i$$ **Final answer:** $ -\frac{29}{53} - \frac{31}{53}i $ --- **Slug:** complex division **Subject:** algebra **Desmos:** {"latex":"y=0","features":{"intercepts":true,"extrema":true}} **q_count:** 1