11.4. Find all natural numbers $x$ such that the product of all digits in the decimal representation of $x$ equals $x^{2}-10 x-22$
Answer: $x=12$.

Solution: First, the product of all digits of a natural number is non-negative, so $x^{2}-10 x-22$, from which $x \geq \frac{10+\sqrt{188}}{2}$, that is, $x \geq 12$. Second, if in the product of all digits of a natural number, all digits except the first are replaced by tens, the product will not decrease, but will not exceed the number itself, so the product of all digits of the number does not exceed the number itself, hence $x^{2}-11 x-22 \leq 0$, from which $x \leq \frac{11+\sqrt{209}}{2}$, that is, $x \leq 12$. For $x=12$, the condition is obviously satisfied, therefore, this is the only answer.

Grading criteria. Only the answer with verification: 1 point. Proof that the product of all digits of the number does not exceed the number itself: 3 points.