How to compute a probability

Question

I have that X ~ N(0, 1) and I want to compute the probability P(X² + X + 1 > 0). Normally, I proceed to reduce the expression into something that looks like P(X > c), but on this problem the roots of X² + X + 1 are not real. Can I proceed using imaginary roots?

Answer 1

A quadratic with no real roots either lies completely above or completely below y=0. In your case, the quadratic with no real roots and positive end behavior will always be greater than zero. So, seems like P = 1, if I have understood correctly.