1. **State the problem:** Simplify the expression $$(2q + \frac{1}{2})(3p + 5q)$$. 2. **Formula used:** Use the distributive property (FOIL method) for multiplying two binomials: $$(a+b)(c+d) = ac + ad + bc + bd$$. 3. **Apply the distributive property:** $$2q \times 3p = 6pq$$ $$2q \times 5q = 10q^2$$ $$\frac{1}{2} \times 3p = \frac{3p}{2}$$ $$\frac{1}{2} \times 5q = \frac{5q}{2}$$ 4. **Combine all terms:** $$6pq + 10q^2 + \frac{3p}{2} + \frac{5q}{2}$$ 5. **Final answer:** $$6pq + 10q^2 + \frac{3p}{2} + \frac{5q}{2}$$ This is the simplified expanded form of the given expression.