Question

How do you factor `x^2+1`?

Answer 1

You can do it using Complex Numbers! A complex number is given as `a+ib` 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:
`(x+i)(x-i)`
Try to multiply them remembering that `i=sqrt(-1)`
You get:
`x^2-ix+ix-i^2=`
but `i^2=[sqrt(-1)]^2=-1` so
`=x^2cancel(-ix)+cancel(ix)-i^2=x^2+1`