Question

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

Answer 1

To factor `x^2+x+1`
use the quadratic formula for roots
`x=(-b+-sqrt(b^2-4ac))/(2a)`

In this case:
`x= (-1+-sqrt(1^2-4(1)(1)))/(2(1))`

`x = -1/2 +-sqrt(-3)/2`

With no Real roots
and thus no factors if we are restricted to Real numbers.

However, if we are allowed to venture into Complex numbers
`x= -1/2 +-sqrt(3)/2i`

and the factors are
`(x+1/2+sqrt(3)/2i)(x+1/2-sqrt(3)/2)`