The number of integer values of k for which 2x^2+kx+8>0 for every real x is

Since the leading coefficient is positive, the quadratic is positive for all real $x$ iff the discriminant is negative: $k^2-64<0$, i.e. $-8<k<8$. The integers are $-7,-6,\ldots,7$, which are $15$ values. Using $|k|\le8$ would add the boundary values where the expression touches zero, giving $17$; strict inequality excludes them.