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. **Rewrite the polynomial:** Combine like terms: $$4x^3 + (a - 6)x + 3$$ 3. **Use Polynomial Remainder Theorem:** The remainder when dividing by $2x - 1$ can be found by evaluating the polynomial at the root of $2x - 1 = 0$. That root is: $$x = \frac{1}{2}$$ 4. **Evaluate the polynomial at $x = \frac{1}{2}$:** $$4\left(\frac{1}{2}\right)^3 + (a - 6)\left(\frac{1}{2}\right) + 3 = 7$$ 5. **Simplify:** $$4\times \frac{1}{8} + \frac{a - 6}{2} + 3 = 7$$ $$\frac{1}{2} + \frac{a - 6}{2} + 3 = 7$$ 6. **Multiply both sides by 2 to clear denominators:** $$1 + a - 6 + 6 = 14$$ (Notice $\frac{a - 6}{2} \times 2 = a - 6$, and also $\frac{1}{2} \times 2 = 1$, $3 \times 2 = 6$.) 7. **Simplify terms:** $$1 + a - 6 + 6 = 14$$ $$1 + a = 14$$ 8. **Solve for $a$:** $$a = 14 - 1 = 13$$ **Final answer:** $$a = 13$$