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 $\displaystyle -1$ is certainly not a primitive root for any prime $\displaystyle p>3$, but $\displaystyle (-1)^{(p-1)/2}=-1$ for primes of the form $\displaystyle p=4n+3$...
What you mean is that if $\displaystyle x$ is a primtive root then $\displaystyle x^{(p-1)/2}=-1$, i.e. if $\displaystyle x$ is a primitive root then it's not a quadratic residue. Since the product of two nonresidues is a residue...