# How do you factor x^2+1?

You can do it using Complex Numbers! A complex number is given as $a + i b$ where $i = \sqrt{- 1}$ represents the Immaginary Unit (notice that $\sqrt{- 1}$ cannot be evaluated as a real number so you call it simply $i$).
In your case you have that, to get ${x}^{2} + 1$, you need to use two complex numbers:
$\left(x + i\right) \left(x - i\right)$
Try to multiply them remembering that $i = \sqrt{- 1}$
${x}^{2} - i x + i x - {i}^{2} =$
but ${i}^{2} = {\left[\sqrt{- 1}\right]}^{2} = - 1$ so
$= {x}^{2} \cancel{- i x} + \cancel{i x} - {i}^{2} = {x}^{2} + 1$