1. **State the problem:** Simplify or solve the expression $$\frac{\sqrt{3}x + 2 + \sqrt{x}}{\sqrt{3}x + 2 - \sqrt{x}}$$. 2. **Identify the structure:** This is a fraction with a numerator and denominator that are similar except for the sign before $\sqrt{x}$. 3. **Use the conjugate to simplify:** Multiply numerator and denominator by the conjugate of the denominator: $$\frac{\sqrt{3}x + 2 + \sqrt{x}}{\sqrt{3}x + 2 - \sqrt{x}} \times \frac{\sqrt{3}x + 2 + \sqrt{x}}{\sqrt{3}x + 2 + \sqrt{x}}$$ 4. **Apply the difference of squares formula:** $$ (a - b)(a + b) = a^2 - b^2 $$ where $$a = \sqrt{3}x + 2, \quad b = \sqrt{x}$$ 5. **Calculate the denominator:** $$ (\sqrt{3}x + 2)^2 - (\sqrt{x})^2 $$ 6. **Expand the square:** $$ (\sqrt{3}x)^2 + 2 \times \sqrt{3}x \times 2 + 2^2 - x $$ $$ = 3x^2 + 4\sqrt{3}x + 4 - x $$ 7. **Simplify the denominator:** $$ 3x^2 + 4\sqrt{3}x + 4 - x $$ 8. **Calculate the numerator:** $$ (\sqrt{3}x + 2 + \sqrt{x})^2 $$ 9. **Expand the numerator:** $$ (\sqrt{3}x)^2 + 2 \times \sqrt{3}x \times 2 + 2^2 + 2 \times \sqrt{3}x \times \sqrt{x} + 2 \times 2 \times \sqrt{x} + (\sqrt{x})^2 $$ $$ = 3x^2 + 4\sqrt{3}x + 4 + 2\sqrt{3}x^{3/2} + 4\sqrt{x} + x $$ 10. **Final simplified form:** $$ \frac{3x^2 + 4\sqrt{3}x + 4 + 2\sqrt{3}x^{3/2} + 4\sqrt{x} + x}{3x^2 + 4\sqrt{3}x + 4 - x} $$ This is the simplified expression after rationalizing the denominator. **Note:** To solve for $x$, you would set this expression equal to a value and solve the resulting equation, which may require numerical methods depending on the context.