1. **State the problem:** Solve the equation $$4\sqrt{x} + 3 = 6\sqrt{x} - 2$$ for $x$. 2. **Isolate the square root term:** Move all terms involving $\sqrt{x}$ to one side and constants to the other. $$4\sqrt{x} + 3 = 6\sqrt{x} - 2$$ Subtract $4\sqrt{x}$ from both sides: $$\cancel{4\sqrt{x}} + 3 = 6\sqrt{x} - \cancel{4\sqrt{x}} - 2$$ which simplifies to $$3 = 2\sqrt{x} - 2$$ 3. Add 2 to both sides to isolate the term with $\sqrt{x}$: $$3 + 2 = 2\sqrt{x} - 2 + 2$$ $$5 = 2\sqrt{x}$$ 4. Divide both sides by 2 to solve for $\sqrt{x}$: $$\frac{5}{\cancel{2}} = \frac{2\sqrt{x}}{\cancel{2}}$$ $$\frac{5}{2} = \sqrt{x}$$ 5. Square both sides to solve for $x$: $$\left(\frac{5}{2}\right)^2 = (\sqrt{x})^2$$ $$\frac{25}{4} = x$$ 6. **Final answer:** $$x = \frac{25}{4}$$