1. **State the problem:** Expand and simplify the expression $$(a - 4x)\left(\frac{1}{2}a + 3x\right)$$. 2. **Formula used:** Use the distributive property (FOIL method) to multiply each term in the first parenthesis by each term in the second parenthesis: $$ (a - 4x)\left(\frac{1}{2}a + 3x\right) = a \cdot \frac{1}{2}a + a \cdot 3x - 4x \cdot \frac{1}{2}a - 4x \cdot 3x $$ 3. **Multiply each term:** $$ = \frac{1}{2}a^2 + 3ax - 2ax - 12x^2 $$ 4. **Combine like terms:** $$ 3ax - 2ax = (3 - 2)ax = 1ax = ax $$ 5. **Final simplified expression:** $$ \frac{1}{2}a^2 + ax - 12x^2 $$ This is the expanded and simplified form of the given expression.