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:** The factorial $n!$ is the product of all positive integers up to $n$. Also, $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.\n\n3. **Simplify the right side:** Since $3! = 3 \times 2 \times 1 = 6$, we have $$\sqrt{3!} \times 7 = \sqrt{6} \times 7 = 7\sqrt{6}.$$\n\n4. **Simplify the left side:** Using factorial properties, $$\frac{(x+2)!}{x!} = (x+2)(x+1)\quad \text{because} \quad (x+2)! = (x+2)(x+1)x!.$$\nSo the left side becomes $$\sqrt{(x+2)(x+1)}.$$\n\n5. **Set the equation:** $$\sqrt{(x+2)(x+1)} = 7\sqrt{6}.$$\n\n6. **Square both sides to remove the square roots:** $$\left(\sqrt{(x+2)(x+1)}\right)^2 = \left(7\sqrt{6}\right)^2,$$\nwhich simplifies to $$ (x+2)(x+1) = 49 \times 6 = 294.$$\n\n7. **Expand the left side:** $$x^2 + 3x + 2 = 294.$$\n\n8. **Bring all terms to one side:** $$x^2 + 3x + 2 - 294 = 0,$$\nwhich simplifies to $$x^2 + 3x - 292 = 0.$$\n\n9. **Solve the quadratic equation:** Using the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=1$, $b=3$, $c=-292$, we get\n$$x = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times (-292)}}{2} = \frac{-3 \pm \sqrt{9 + 1168}}{2} = \frac{-3 \pm \sqrt{1177}}{2}.$$\n\n10. **Evaluate $\sqrt{1177}$:** Approximately $34.31$, so\n$$x = \frac{-3 \pm 34.31}{2}.$$\n\n11. **Find the two possible solutions:**\n- $$x = \frac{-3 + 34.31}{2} = \frac{31.31}{2} = 15.655,$$\n- $$x = \frac{-3 - 34.31}{2} = \frac{-37.31}{2} = -18.655.$$\n\n12. **Check for valid factorial domain:** Factorials are defined for non-negative integers, so $x$ must be a non-negative integer. The only valid integer near $15.655$ is $16$.\n\n13. **Verify $x=16$:**\nCalculate $(x+2)(x+1) = 18 \times 17 = 306$, which is not equal to 294, so $x=16$ is not exact.\n\n14. **Check $x=15$:**\n$(15+2)(15+1) = 17 \times 16 = 272$, not 294.\n\n15. **Check $x=14$:**\n$(14+2)(14+1) = 16 \times 15 = 240$, no.\n\n16. **Check $x=13$:**\n$(13+2)(13+1) = 15 \times 14 = 210$, no.\n\n17. **Check $x=17$:**\n$(17+2)(17+1) = 19 \times 18 = 342$, no.\n\n18. **Since no integer $x$ satisfies exactly, the solution is $x = \frac{-3 \pm \sqrt{1177}}{2}$, but factorials require integer $x$.**\n\n**Final answer:** $$x = \frac{-3 \pm \sqrt{1177}}{2}.$$