[b]9.[/b] Find all pairs of linear polynomials $f(x)$, $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$, $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$. [b](A. 8)[/b]
**
   This equation has a solution if and only if the quotient ring \( \mathbb{Z}[x]/(s, ex + t) \) is zero. This quotient is zero if and only if:
   \[
   s \mid e \quad \text{and} \quad \gcd(t, s) = 1
   \]

6. **Final Conditions:**
   Notice that:
   \[
   \begin{pmatrix} q & -p \\ \alpha & \beta \end{pmatrix} \begin{pmatrix} b \\ d \end{pmatrix} = \begin{pmatrix} s \\ t \end{pmatrix}
   \]
   Therefore, \( \gcd(t, s) = 1 \) if and only if \( \gcd(b, d) = 1 \).

7. **Conclusion:**
   The original equation \( f(x)u(x) + g(x)v(x) = 1 \) has a solution if and only if:
   \[
   bc - ad \mid \gcd(a, c)^2 \quad \text{and} \quad \gcd(b, d) = 1
   \]

The final answer is \( \boxed{bc - ad \mid \gcd(a, c)^2 \text{ and } \gcd(b, d) = 1} \)