I've been struggling to find a way to solve this question for a while now.

I can prove that there is at least one real root using IVT but have no idea how to prove that there can exist more.

Currently my idea is to prove that there cannot be $5$ real roots somehow and then I can say that as complex numbers come with conjugates, there cannot only be one complex root and so there can be at most $3$ real roots.

I had another idea to assume that there are at most $2$ real roots. Then use the fact that complex conjugates come in pairs to say there must be at most 3 roots as we have already shown there are at least $2$. But then there would be nothing to account for if there are $5$ real roots.

Am I just overthinking this and there's an easier way to go about it?