1. **Stating the problem:** We want to understand why the inequality $$gd \leq (1+k) V^2$$ holds, starting from the given inequality involving $$g^2 d^2$$ and $$V^4$$. 2. **Given inequality:** $$g^2 d^2 \leq (k^2 - 1) V^4 + 2 g d V^2$$ 3. **Goal:** Rearrange this inequality to isolate $$gd$$ and find a simpler upper bound. 4. **Rewrite the inequality:** Move all terms to one side: $$g^2 d^2 - 2 g d V^2 - (k^2 - 1) V^4 \leq 0$$ 5. **Consider this as a quadratic inequality in $$gd$$:** Let $$x = gd$$, then $$x^2 - 2 V^2 x - (k^2 - 1) V^4 \leq 0$$ 6. **Solve the quadratic inequality:** The quadratic equation is $$x^2 - 2 V^2 x - (k^2 - 1) V^4 = 0$$ 7. **Find roots using the quadratic formula:** $$x = \frac{2 V^2 \pm \sqrt{(2 V^2)^2 + 4 (k^2 - 1) V^4}}{2}$$ Simplify inside the square root: $$= \frac{2 V^2 \pm \sqrt{4 V^4 + 4 (k^2 - 1) V^4}}{2} = \frac{2 V^2 \pm \sqrt{4 V^4 (1 + k^2 - 1)}}{2} = \frac{2 V^2 \pm \sqrt{4 k^2 V^4}}{2}$$ 8. **Simplify the square root:** $$= \frac{2 V^2 \pm 2 k V^2}{2} = V^2 (1 \pm k)$$ 9. **Roots are:** $$x_1 = V^2 (1 - k), \quad x_2 = V^2 (1 + k)$$ 10. **Since the quadratic opens upward (coefficient of $$x^2$$ is positive), the inequality $$x^2 - 2 V^2 x - (k^2 - 1) V^4 \leq 0$$ holds between the roots:** $$V^2 (1 - k) \leq gd \leq V^2 (1 + k)$$ 11. **Assuming $$gd$$ is positive and $$k \geq 0$$, the lower bound is less relevant, so the key inequality is:** $$gd \leq (1 + k) V^2$$ 12. **This matches the boxed inequality and shows the upper bound on $$gd$$ derived from the original inequality.**