3. Find all pairs of real numbers $a$ and $b$, for which there exists a polynomial $f(x)$ such that $f(f(x))=\left(x^{2}+a x+b\right)^{2}$.
Answer: $a$ - arbitrary, $b=\frac{a}{2}+\frac{a^{2}}{4}$.

## Solution (outline)

The polynomial has the form (this needs to be explained) $f(x)=x^{2}+p x+q$. By expanding both sides of the equation, we arrive at four equations involving $p$ and $q$ with parameters $a$ and $b$. This system must be solvable, from which we obtain the conditions: $a$ - arbitrary, $b=\frac{a}{2}+\frac{a^{2}}{4}$.

## Criteria:

|  | fully ar |
| :---: | :---: |
| 6  $5-4$  4 | The answer is correct, but the solution is not fully justified. |
|  | The answer is incorrect, with one or two arithmetic errors. |
| 3 | The answer is incorrect, with an error in the application of formulas for multiplication or the idea of the solution without an answer. |
| $2-1$ | There are minor advancements in the solution. Special cases are correctly considered. |