Doubt in finding number of non-prime factors of an integer

You are missing making the error that your module made. You are right, the module is wrong.

One could speculate about why the person hired to solve the problems made the mistake. But note that $$11 \times 4 \times 10=440$$ so the person doing the solutions may not have noticed that $4$ is not prime. You did notice.

Your argument looks right to me. I think there is a misprint - either the $4$ in $4^{10}$ was supposed to be a prime number (probably $3$, given that $5$ was already in the factorization), or the $4$ in $437$ was supposed to be an $8$.