1. **State the problem:** Simplify the expression $$\left(\frac{1}{2}a + \frac{2}{3}\right)\left(4a - \frac{3}{2}b\right)$$. 2. **Recall the distributive property:** To multiply two binomials, use the FOIL method (First, Outer, Inner, Last). 3. **Apply FOIL:** - First: $$\frac{1}{2}a \times 4a = 2a^2$$ - Outer: $$\frac{1}{2}a \times \left(-\frac{3}{2}b\right) = -\frac{3}{4}ab$$ - Inner: $$\frac{2}{3} \times 4a = \frac{8}{3}a$$ - Last: $$\frac{2}{3} \times \left(-\frac{3}{2}b\right) = -1b = -b$$ 4. **Combine all terms:** $$2a^2 - \frac{3}{4}ab + \frac{8}{3}a - b$$ 5. **Final simplified expression:** $$\boxed{2a^2 - \frac{3}{4}ab + \frac{8}{3}a - b}$$