<s>[INST] Number of possible integer values of $x$ for which $\frac{x^3+2x^2+9}{x^2+4x+5}$ is integer How many integer numbers, $x$, verify that the following
\begin{equation*}
\frac{x^3+2x^2+9}{x^2+4x+5}
\end{equation*}
is an integer?
I managed to do:
\begin{equation*}
\frac{x^3+2x^2+9}{x^2+4x+5} = x-2 + \frac{3x+19}{x^2+4x+5}
\end{equation*}
but I cannot go forward.
 [/INST] You're off to a good start. Now note that the denominator $x^2+4x+5$ is quickly larger than the numerator $3x+19$; you can quickly reduce the problem to only finitely many values for $x$ to check.
More details: (Hover to show)

 The fraction is certainly not an integer if the denominator is greater than the numerator, i.e. if $$x^2+4x+5>3x+19,$$ unless perhaps the numerator is zero, but that is not possible in this case. The quadratic formula shows that the inequality above holds if $x\leq5$ or $x\geq4$. Then it remains to check whether the fraction is an integer for $x$ in the range $-4\leq x\leq3$.

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