4. Given the function $f(x)=\frac{2 x^{3}-6 x^{2}+13 x+10}{2 x^{2}-9 x}$

Determine all positive integers $x$ for which $f(x)$ is an integer.
4. For integer $x$, the numerator and the denominator of the fraction are both integers. For $f(x)$ to be an integer, the denominator must be a divisor of the numerator. Since $x$ is a divisor of the denominator and also of part of the numerator, namely $2 x^{3}-6 x^{2}+13 x$, $x$ must be a divisor of 10. Therefore, we only need to try the positive divisors of 10, which are 1, 2, 5, and 10 itself. Substitution yields $f(5)=35$ and $f(10)=14$.