1. The problem states that when the polynomial $$4x^3 - 6x + ax + 3$$ is divided by $$2x - 1$$, the remainder is 7. 2. By the Remainder Theorem, the remainder of a polynomial $$f(x)$$ when divided by $$2x - 1$$ is equal to $$f\left(\frac{1}{2}\right)$$. 3. We substitute $$x = \frac{1}{2}$$ into the polynomial: $$f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 6\left(\frac{1}{2}\right) + a\left(\frac{1}{2}\right) + 3$$ 4. Simplify each term: $$4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$$ $$-6 \times \frac{1}{2} = -3$$ $$a \times \frac{1}{2} = \frac{a}{2}$$ 5. So, $$f\left(\frac{1}{2}\right) = \frac{1}{2} - 3 + \frac{a}{2} + 3 = \frac{1}{2} + \frac{a}{2}$$ 6. According to the problem, this value must be equal to 7: $$\frac{1}{2} + \frac{a}{2} = 7$$ 7. Subtract $$\frac{1}{2}$$ from both sides: $$\frac{a}{2} = 7 - \frac{1}{2} = \frac{14}{2} - \frac{1}{2} = \frac{13}{2}$$ 8. Multiply both sides by 2: $$a = 13$$ **Final answer:** $$a = 13$$