C/C++ :: Check If Quartic Has Real Roots?

I have been writing code to find the roots of a quartic (with coefficients input by the user) by the Newton Rapson method. The method fails if no real roots exist so I have been trying to implement an initial check to see whether the equation has real roots. I've been trying to use the intermediate value theorem but it only works for functions with a positive and negative tail. extend the bounds (fa and fb) to check over many ranges or any other mathematical method of better effect?

Atm I define fa<0<fb where fa=F(-10) and fb=F(10).

#include <stdio.h>
#include <math.h>
#include <conio.h>
#include <stdlib.h>
int a0, a1, a2, a3, a4;
double F(double x) //Sets main function {
return (a0+a1*x+a2*pow(x, 2)+a3*pow(x, 3)+a4*pow(x, 4));

[code].....