1. **State the problem:** We are given the polynomial $$4x^3 - 6x + ax + 3$$ and told that when it is divided by $$2x - 1$$, the remainder is 7. 2. **Understand the divisor:** First, we set $$2x - 1 = 0$$ to find the value of $$x$$ at which the remainder is evaluated. From $$2x - 1 = 0$$, we get $$x = \frac{1}{2}$$. 3. **Apply the Remainder Theorem:** The remainder when a polynomial $$f(x)$$ is divided by $$x - c$$ is $$f(c)$$. Since the divisor is $$2x - 1$$, replace $$x$$ with $$\frac{1}{2}$$ in the polynomial and set the expression equal to 7: $$4\left(\frac{1}{2}\right)^3 - 6\left(\frac{1}{2}\right) + a\left(\frac{1}{2}\right) + 3 = 7$$ 4. **Simplify each term:** $$4\left(\frac{1}{8}\right) - 3 + \frac{a}{2} + 3 = 7$$ $$\frac{4}{8} - 3 + \frac{a}{2} + 3 = 7$$ $$\frac{1}{2} + \frac{a}{2} = 7$$ 5. **Combine like terms:** $$\frac{1}{2} + \frac{a}{2} = 7$$ Multiply both sides by 2 to clear the denominator: $$1 + a = 14$$ 6. **Solve for $$a$$:** $$a = 14 - 1 = 13$$ **Final answer:** $$a = 13$$