1. **State the problem:** We are given the equation $$\sqrt{\frac{2x + 3}{2}} = y + 1$$ and need to express $x$ in terms of $y$. 2. **Isolate and square both sides:** To eliminate the square root, square both sides: $$\left(\sqrt{\frac{2x + 3}{2}}\right)^2 = (y + 1)^2$$ which simplifies to $$\frac{2x + 3}{2} = (y + 1)^2$$ 3. **Multiply both sides by 2:** $$2x + 3 = 2(y + 1)^2$$ 4. **Expand the right side:** $$(y + 1)^2 = y^2 + 2y + 1$$ so $$2x + 3 = 2(y^2 + 2y + 1) = 2y^2 + 4y + 2$$ 5. **Solve for $x$:** $$2x = 2y^2 + 4y + 2 - 3 = 2y^2 + 4y - 1$$ $$x = \frac{2y^2 + 4y - 1}{2}$$ 6. **Conclusion:** The expression for $x$ in terms of $y$ is $$x = \frac{2y^2 + 4y - 1}{2}$$ This corresponds to option B.