1. The problem is to rearrange the equation $$2y = \sqrt{x + 3} + 2$$ to make $$x$$ the subject. 2. Start by isolating the square root term on one side: $$2y - 2 = \sqrt{x + 3}$$ 3. Square both sides to eliminate the square root: $$(2y - 2)^2 = (\sqrt{x + 3})^2$$ $$ (2y - 2)^2 = x + 3 $$ 4. Expand the left side: $$ (2y - 2)^2 = (2y)^2 - 2 \times 2y \times 2 + 2^2 = 4y^2 - 8y + 4 $$ 5. Substitute back: $$4y^2 - 8y + 4 = x + 3$$ 6. Solve for $$x$$ by subtracting 3 from both sides: $$x = 4y^2 - 8y + 4 - 3$$ $$x = 4y^2 - 8y + 1$$ Final answer: $$\boxed{x = 4y^2 - 8y + 1}$$