Existence of Real Polynomial with no Real Root

Theorem
There exist polynomials in real numbers $\R$ which have no roots in $\R$.

Proof
Proof by Counterexample

Take the quadratic equation:
 * $(1): \quad x^2 + 1 = 0$

From the Quadratic Formula, the solution to $(1)$ is:

But there is no real number $x$ such that:
 * $x^2 = -1$

Hence the result.