How do I prove a quadratic expression to always be negative for all real values of x

Question

So in school we are learning about quadratics. But I'm very confused on how to prove the question above.

Example: $-x^{2}+x-2$

How would I prove that the expression is negative for all real values of x for expression above.

Answer 1

$$-x^2+x-2=\dfrac{4x^2-4x+8}{-4}=\dfrac{(2x-1)^2+7}{-4}$$

Now, $(2x-1)^2+7\ge7$ for real $x$

Answer 2

A general criterion:

The discriminant is negative (hence no real root, hence constant sign) and the leading coefficient is negative. Or the constant term is negative.