1. **State the problem:** Solve for $x$ in the equation $$\sqrt{\frac{(x+2)!}{x!}} = \sqrt{3!} \times 7.$$\n\n2. **Recall factorial and square root properties:**\n- Factorial: $n! = n \times (n-1) \times \cdots \times 1$.\n- Square root of a quotient: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.\n\n3. **Simplify the left side:**\n$$\sqrt{\frac{(x+2)!}{x!}} = \sqrt{(x+2)(x+1) \cancel{\frac{x!}{x!}}} = \sqrt{(x+2)(x+1)}.$$\n\n4. **Simplify the right side:**\n$$\sqrt{3!} \times 7 = \sqrt{6} \times 7 = 7\sqrt{6}.$$\n\n5. **Set the simplified expressions equal:**\n$$\sqrt{(x+2)(x+1)} = 7\sqrt{6}.$$\n\n6. **Square both sides to remove the square roots:**\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. **Form 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 $a=1$, $b=3$, $c=-292$.\n\nCalculate the discriminant:\n$$\Delta = 3^2 - 4 \times 1 \times (-292) = 9 + 1168 = 1177.$$\n\n11. **Find the roots:**\n$$x = \frac{-3 \pm \sqrt{1177}}{2}.$$\n\n12. **Check for integer solutions:**\nSince factorials are defined for non-negative integers, $x$ must be a non-negative integer.\n\nApproximate $\sqrt{1177} \approx 34.31$, so:\n$$x_1 = \frac{-3 + 34.31}{2} \approx 15.65,$$\n$$x_2 = \frac{-3 - 34.31}{2} \approx -18.65.$$\n\nOnly $x=15$ is close to an integer and non-negative.\n\n13. **Verify $x=15$:**\n$$\sqrt{\frac{17!}{15!}} = \sqrt{16 \times 17} = \sqrt{272} \approx 16.49,$$\n$$7\sqrt{6} \approx 7 \times 2.45 = 17.15,$$\nwhich is close but not exact.\n\nTry $x=14$:\n$$\sqrt{\frac{16!}{14!}} = \sqrt{15 \times 16} = \sqrt{240} \approx 15.49,$$\nless than $17.15$.\n\nTry $x=13$:\n$$\sqrt{14 \times 15} = \sqrt{210} \approx 14.49,$$\nless than $17.15$.\n\nTry $x=16$:\n$$\sqrt{18 \times 17} = \sqrt{306} \approx 17.49,$$\nslightly more than $17.15$.\n\nSince $x$ must be integer and the exact solution is not integer, the exact solution is $$x = \frac{-3 + \sqrt{1177}}{2}.$$\n\n**Final answer:** $$x = \frac{-3 + \sqrt{1177}}{2}.$$