4. Find all natural numbers $a$ for which the number

$$
\frac{a+1+\sqrt{a^{5}+2 a^{2}+1}}{a^{2}+1}
$$

is also a natural number.
Answer. $a=1$.

Solution. Let $a+1=b, \sqrt{a^{5}+2 a^{2}+1}=c$. Write the polynomial $c^{2}-b^{2}=$ $a^{5}+2 a^{2}+1-(a+1)^{2}=a^{5}+a^{2}-2 a$, and divide it by $a^{2}+1$ using long division, we get a remainder of $-(a+1)$. For $a>1$, the absolute value of the remainder is less than $a^{2}+1$, so the remainder cannot be divisible by $a^{2}+1$ for any $a>1$. If $c^{2}-b^{2}$ is not divisible by the denominator $a^{2}+1$, then $b+c$ is also not divisible. The equation is satisfied by the unique value $a=1$.

Comment. Correct solution - 20 points. There is a gap in the justification - 15 points. The value $a=1$ is found, but it is not proven that there are no other solutions - 1 point. The proof is started - another 1 point. The proof is started, and there is some progress - 5 points. The proof is partially conducted - 10 points.