The greatest common divisor of the numerator and the denominator

$$
9 x^{3}-18 x^{2}-16 x+32
$$

when simplified with which expression, results in:

$$
\frac{2 x+3}{2 x-3}
$$

The problem was solved by: Füstös P., Harsányi Z., Jánossy Gy., Makó E., Paunz A., Petrik Sarolta, Reich Z., Sárközy E., Schlesinger Gy., Steiger J., Szabó J., Tóth B. - The 5th grade of the Budapest branch of the Evangelical Gymnasium.
Let's simplify the following fraction:

$$
\frac{18 x^{4}-9 x^{3}-86 x^{2}+16 x+96}{18 x^{4}-63 x^{3}+22 x^{2}+112 x-96}
$$