For what primes p (not 2 or 5) do we have
(10/p) = 1?
$\displaystyle \left(\frac{10}{p}\right) = \left(\frac{2}{p}\right) \left(\frac{5}{p}\right) $
So if $\displaystyle \left(\frac{10}{p}\right)=1 $, then $\displaystyle \left(\frac{2}{p}\right)=\left(\frac{5}{p}\right)= 1 $ or $\displaystyle \left(\frac{2}{p}\right)=\left(\frac{5}{p}\right)=-1 $
Now you can use quadratic reciprocity to find $\displaystyle \left(\frac{2}{p}\right) $ and $\displaystyle \left(\frac{5}{p}\right) $.