Example 4 Try to determine all positive integer pairs $(a, b)$ such that $a b^{2}+b+7$ divides $a^{2} b+$ $a+b$.
(IMO 38th problem)
Analysis and Solution: According to the problem, the quotient of $a^{2} b+a+b$ divided by $a b^{2} + b + 7$ is an integer. If the division operation's quotient and remainder can both be expressed in terms of $a$ and $b$, the problem would be easier to solve. However, this is unlikely because both the divisor and the dividend are cubic polynomials in $a$ and $b$. Further consideration reveals that if we can reduce the degree of the dividend, we can apply the conclusion from the commentary of Example 1 to solve it. After some attempts, we discovered the following identity to eliminate the cubic terms:
$$
\begin{array}{l}
b\left(a^{2} b+a+b\right)-a\left(a b^{2}+b+7\right) \\
=b^{2}-7 a .
\end{array}
$$

Thus, according to the problem, $a b^{2} + b + 7$ divides $b^{2} - 7 a$. At this point, the dividend is a quadratic polynomial.
Next, we need to make some precise estimates in different cases.
(i) If $b^{2} - 7 a > 0$, then it must hold that
$$
a b^{2} + b + 7 \leqslant b^{2} - 7 a < b^{2}.
$$

But this is impossible.
$$
\begin{array}{l}
\text { (ii) If } b^{2} - 7 a < 0, \text { then } \\
a b^{2} + b + 7 \leqslant 7 a - b^{2} < 7 a .
\end{array}
$$

Thus, $b^{2} < 7$, which means $b = 1$ or $2$.
When $b = 1$, the condition of the problem becomes $a + 8$ divides $a^{2} + a + 1$. Noting that $a^{2} + a + 1 = (a + 8)(a - 7) + 57$, it follows that $a + 8$ must be a divisor of $57 = 3 \times 19$, i.e.,
$a = 19 - 8 = 11$ or $57 - 8 = 49$.
When $b = 2$, we have $4a + 9$ divides $4 - 7a$. Since $0 < 7a - 4 < 2(4a + 9)$, and $4a + 9 = 7a - 4$ has no integer solutions, this case is impossible.
(iii) If $b^{2} - 7a = 0$, then $b = 7k$ and $a = 7k^{2}$, where $k$ is a positive integer. In this case, the quotient of $a^{2} b + a + b$ divided by $a b^{2} + b + 7$ is exactly $k$, and the condition of the problem is satisfied.

In summary, all pairs of positive integers $(a, b)$ that satisfy the problem's requirements are
$$
(11,1),(49,1),\left(7 k^{2}, 7 k\right)(k \in \mathbf{N}) \text {. }
$$