Can someone please help me with this?

Given that the roots of the equation $\displaystyle x^2+ax+(a+2)=0$ differ by two, find the possible values of the constant a?

I used the symmetry axis of the parabola $\displaystyle \frac{-b}{2a}=1$to find one of the possible values =-2, but my teacher said, when we were checking this that 6 is also a possible answer. How is this possible? Is my method of solving this correct?

Thank you!