1. **State the problem:** Simplify the complex fraction $$\frac{7 - 17i}{2 - 3i}$$ into the form $a + bi$ where $a$ and $b$ are real numbers. 2. **Formula and rule:** To simplify a complex fraction, multiply numerator and denominator by the conjugate of the denominator. The conjugate of $2 - 3i$ is $2 + 3i$. 3. **Multiply numerator and denominator:** $$\frac{7 - 17i}{2 - 3i} \times \frac{2 + 3i}{2 + 3i} = \frac{(7 - 17i)(2 + 3i)}{(2 - 3i)(2 + 3i)}$$ 4. **Expand numerator:** $$ (7)(2) + (7)(3i) - (17i)(2) - (17i)(3i) = 14 + 21i - 34i - 51i^2 $$ 5. **Simplify numerator using $i^2 = -1$:** $$ 14 + 21i - 34i - 51(-1) = 14 - 13i + 51 = 65 - 13i $$ 6. **Expand denominator:** $$ (2)(2) + (2)(3i) - (3i)(2) - (3i)(3i) = 4 + 6i - 6i - 9i^2 $$ 7. **Simplify denominator:** $$ 4 + 0 - 9(-1) = 4 + 9 = 13 $$ 8. **Write the fraction:** $$ \frac{65 - 13i}{13} $$ 9. **Simplify by dividing numerator and denominator by 13:** $$ \frac{\cancel{13} \times 5 - \cancel{13} \times i}{\cancel{13}} = 5 - i $$ 10. **Final answer:** The simplified form is $5 - i$.