1. **Problem:** Write $\frac{3 + 2i}{4 - 3i}$ in the form $a + bi$. 2. **Formula and rules:** To write a complex fraction in the form $a + bi$, multiply numerator and denominator by the conjugate of the denominator. 3. **Step-by-step solution:** Multiply numerator and denominator by the conjugate of the denominator $4 + 3i$: $$\frac{3 + 2i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i} = \frac{(3 + 2i)(4 + 3i)}{(4 - 3i)(4 + 3i)}$$ Calculate numerator: $$(3)(4) + (3)(3i) + (2i)(4) + (2i)(3i) = 12 + 9i + 8i + 6i^2$$ Recall $i^2 = -1$, so: $$12 + 9i + 8i + 6(-1) = 12 + 17i - 6 = 6 + 17i$$ Calculate denominator: $$(4)^2 - (3i)^2 = 16 - 9i^2 = 16 - 9(-1) = 16 + 9 = 25$$ Divide numerator by denominator: $$\frac{6 + 17i}{25} = \frac{6}{25} + \frac{17}{25}i$$ 4. **Final answer:** $$\boxed{\frac{6}{25} + \frac{17}{25}i}$$ This is the expression in the form $a + bi$ where $a = \frac{6}{25}$ and $b = \frac{17}{25}$.