1. **State the problem:** Write the expression $$\frac{3 + i}{2 - i} + \frac{4 + 10i}{-9 + 7i}$$ in the form $$a + ib$$ where $$a$$ and $$b$$ are real numbers. 2. **Recall the formula:** To write a complex fraction $$\frac{z_1}{z_2}$$ in standard form, multiply numerator and denominator by the conjugate of the denominator: $$\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}}$$ where $$\overline{z_2}$$ is the conjugate of $$z_2$$. 3. **First fraction:** $$\frac{3 + i}{2 - i}$$ - Conjugate of denominator: $$2 + i$$ - Multiply numerator and denominator: $$\frac{(3 + i)(2 + i)}{(2 - i)(2 + i)}$$ - Calculate numerator: $$3 \times 2 + 3 \times i + i \times 2 + i \times i = 6 + 3i + 2i + i^2 = 6 + 5i - 1 = 5 + 5i$$ - Calculate denominator: $$2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5$$ - So first fraction is: $$\frac{5 + 5i}{5} = 1 + i$$ 4. **Second fraction:** $$\frac{4 + 10i}{-9 + 7i}$$ - Conjugate of denominator: $$-9 - 7i$$ - Multiply numerator and denominator: $$\frac{(4 + 10i)(-9 - 7i)}{(-9 + 7i)(-9 - 7i)}$$ - Calculate numerator: $$4 \times (-9) + 4 \times (-7i) + 10i \times (-9) + 10i \times (-7i) = -36 - 28i - 90i - 70i^2 = -36 - 118i + 70 = 34 - 118i$$ - Calculate denominator: $$(-9)^2 - (7i)^2 = 81 - (-49) = 81 + 49 = 130$$ - So second fraction is: $$\frac{34 - 118i}{130} = \frac{34}{130} - \frac{118}{130}i = \frac{17}{65} - \frac{59}{65}i$$ 5. **Add the two results:** $$\left(1 + i\right) + \left(\frac{17}{65} - \frac{59}{65}i\right) = \left(1 + \frac{17}{65}\right) + \left(1 - \frac{59}{65}\right)i = \frac{65}{65} + \frac{17}{65} + \left(\frac{65}{65} - \frac{59}{65}\right)i = \frac{82}{65} + \frac{6}{65}i$$ 6. **Final answer:** $$\boxed{\frac{82}{65} + \frac{6}{65}i}$$