1. **State the problem:** Solve for $x$ in the equation $$\sqrt{\frac{(x+2)!}{x!}} = \sqrt{3!} \times 7.$$\n\n2. **Recall factorial properties:** The factorial function $n!$ is the product of all positive integers up to $n$. Also, note that $$\frac{(x+2)!}{x!} = (x+2)(x+1)$$ because $$\frac{(x+2)!}{x!} = \frac{(x+2)(x+1)x!}{x!} = (x+2)(x+1).$$\n\n3. **Rewrite the equation using this simplification:**\n$$\sqrt{(x+2)(x+1)} = \sqrt{3!} \times 7.$$\n\n4. **Calculate $3!$:**\n$$3! = 3 \times 2 \times 1 = 6.$$\n\n5. **Substitute $3!$ value:**\n$$\sqrt{(x+2)(x+1)} = \sqrt{6} \times 7 = 7\sqrt{6}.$$\n\n6. **Square both sides to eliminate the square root:**\n$$\left(\sqrt{(x+2)(x+1)}\right)^2 = \left(7\sqrt{6}\right)^2$$\n$$ (x+2)(x+1) = 49 \times 6 = 294.$$\n\n7. **Expand the left side:**\n$$x^2 + x + 2x + 2 = x^2 + 3x + 2.$$\n\n8. **Set up the quadratic equation:**\n$$x^2 + 3x + 2 = 294.$$\n\n9. **Bring all terms to one side:**\n$$x^2 + 3x + 2 - 294 = 0$$\n$$x^2 + 3x - 292 = 0.$$\n\n10. **Solve the quadratic equation using the quadratic formula:**\n$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$\nwhere $a=1$, $b=3$, and $c=-292$.\n\n11. **Calculate the discriminant:**\n$$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times (-292) = 9 + 1168 = 1177.$$\n\n12. **Find the roots:**\n$$x = \frac{-3 \pm \sqrt{1177}}{2}.$$\n\n13. **Approximate $\sqrt{1177}$:**\n$$\sqrt{1177} \approx 34.31.$$\n\n14. **Calculate the two possible values for $x$:**\n$$x_1 = \frac{-3 + 34.31}{2} = \frac{31.31}{2} = 15.655,$$\n$$x_2 = \frac{-3 - 34.31}{2} = \frac{-37.31}{2} = -18.655.$$\n\n15. **Check domain restrictions:** Factorials are defined for non-negative integers, so $x$ must be a non-negative integer. Therefore, $x \approx 15.655$ is not an integer, and $x = -18.655$ is negative and invalid.\n\n16. **Check integer values near 15.655:** Try $x=15$ and $x=16$.\n- For $x=15$: $(15+2)(15+1) = 17 \times 16 = 272$ (not 294).\n- For $x=16$: $(16+2)(16+1) = 18 \times 17 = 306$ (not 294).\n\n17. **Conclusion:** There is no integer $x$ satisfying the equation exactly. The problem likely assumes $x$ is a real number, so the solution is $$x = \frac{-3 + \sqrt{1177}}{2} \approx 15.655.$$