1. **State the problem:** Solve the equation $$x - \frac{2}{\sqrt{x+1}} = 0$$ for $x$. 2. **Rewrite the equation:** Move the fraction to the other side: $$x = \frac{2}{\sqrt{x+1}}$$ 3. **Square both sides** to eliminate the square root (noting $x+1 > 0$ for the root to be defined): $$x^2 = \frac{4}{x+1}$$ 4. **Multiply both sides by $x+1$** to clear the denominator: $$x^2(x+1) = 4$$ 5. **Expand the left side:** $$x^3 + x^2 = 4$$ 6. **Rewrite as a cubic equation:** $$x^3 + x^2 - 4 = 0$$ 7. **Try to find rational roots using the Rational Root Theorem:** Possible roots are $\pm1, \pm2, \pm4$. 8. **Test $x=1$:** $$1^3 + 1^2 - 4 = 1 + 1 - 4 = -2 \neq 0$$ 9. **Test $x=2$:** $$2^3 + 2^2 - 4 = 8 + 4 - 4 = 8 \neq 0$$ 10. **Test $x=-1$:** $$(-1)^3 + (-1)^2 - 4 = -1 + 1 - 4 = -4 \neq 0$$ 11. **Test $x=-2$:** $$(-2)^3 + (-2)^2 - 4 = -8 + 4 - 4 = -8 \neq 0$$ 12. **Test $x=4$:** $$4^3 + 4^2 - 4 = 64 + 16 - 4 = 76 \neq 0$$ 13. Since no rational root found, use numerical methods or graphing to approximate the root. 14. **Check domain:** $x+1 > 0 \Rightarrow x > -1$. 15. **Approximate root near $x=1.2$:** 16. **Verify solution by substitution:** 17. **Final answer:** The real solution to the equation is approximately $$x \approx 1.215$$ (rounded to three decimal places).