Show that if g and h are primitive roots of an odd prime p, then their product gh is not a primitive root of p.
That is not true. For instance is certainly not a primitive root for any prime , but for primes of the form ...
What you mean is that if is a primtive root then , i.e. if is a primitive root then it's not a quadratic residue. Since the product of two nonresidues is a residue...