It's just the quadratic equation: \$\frac{-b\pm\sqrt{b^2-4ac}}{2a}\$. In your case, \$a=1\$, \$b=1\$, and \$c=2\$.
So \$\frac{-1\pm\sqrt{-7}}{2}\$ or \$-\frac12\pm j\frac12\sqrt{7}\$.
So \$s^2+s+2=>\left(s-\left[-\frac12+ j\frac12\sqrt{7}\right]\right)\cdot\left(s-\left[-\frac12- j\frac12\sqrt{7}\right]\right)\$.
In freely available Python/SymPy/SageMath:
list(roots(s**2+s+2,s))[-1/2 - sqrt(7)*I/2, -1/2 + sqrt(7)*I/2]expand(prod([s-i for i in list(roots(s**2+s+2,s))]))s**2 + s + 2