l've already read (and understood) a different way to solve this problem, but I wanted to know if there's any way the following idea could work:
Let's assume $P_3P_5$ is a subgroup, then its order should be $3\cdot 5=15$. Therefore, its index would be $2$, so we could conclude it's a normal subgroup. The problem is, $P_3P_5$ would only be a subgroup if either $P_3$ or $P_5$ are normal, so I'm not sure any of this was useful.
Is there any way we can turn this into a working argument?