Applying the quadratic formula, we get
\[ x=\dfrac{-0\pm\sqrt{0^2-4}}{2} =\dfrac{\pm2\,i}{2}=\pm i\]
Using abbreviated synthetic division with \(\pm i\) we get
| \(1\) | \(0\) | \(1\) | |
| \(i\) | \(1\) | \(i\) | \(0\) |
| \(-i\) | \(1\) | \(-i\) | \(0\) |
Which also shows us that \(x^2+1=(x-i)(x+i)\).