How to know if a quadratic is irreducible?

You know if it's not reducible when there isn't any way to factor it using the grouping method. However, you should try to use the quadratic formula to find the roots after attempting to factor. If it still doesn't have any roots, that means the quadratic's vertex is either above or below the x-axis and is facing the other way. (meaning that it never touches the X-Axis)