Prove or Disprove that there exists a real number x such that x^2+x+1 is less than or equal to zero.
From the quadratic equation, you know there are no real roots, meaning the function is never 0 (either always negative, or always positive). With that, you can note that letting x=0 gives you a value of 1, so the function is always positive.
Not quite the most rigorous proof and probably not what your professor is looking for, but it is valid.
Edit: Also, I forgot to add that this works because the function is continuous over all real numbers.
which is one of the criterion of your proof. So the existence of x proves that the function can be less than or equal to 0.
Likely the best method is the intermediate value theorem, if you can do it that way.