Yay ! It is positive!

$\frac{c^{6}-3}{c^{2}+2} = \frac{c^4(c^{2}+2)-2c^{4}-3}{c^2+2} = c^4 - \frac{2c^4+3}{c^2+2} = c^4 - \frac{2c^2(c^2+2)-4c^2+3}{c^2+2} = c^4 - 2c^2 - \frac{-4(c^2+2)+8+3}{c^2+2} = c^4 - 2c^2 +4 -\frac{11}{c^2+2}$

The above expression is an integer if and only if $c^2+2$ divides $11$ . Therefore the only option is $c=3$ .