For any $x \in \mathbb{R}$, it holds that $x^2 + 2x + 5 \neq 0$.

Thus, the negation of the given proposition is $\boxed{\text{For any } x \in \mathbb{R}, x^2 + 2x + 5 \neq 0}$.

To negate the proposition "There exists $x \in \mathbb{R}$ such that $x^{2}+2x+5=0$", we recognize that it is an existential statement. The negation of an existential statement is a universal statement, which means that the claim is false for all elements in the domain.

Hence, the negation of the given proposition is: "For all $x \in \mathbb{R}$, it holds that $x^{2}+2x+5 \neq 0$."

Let's strengthen the proof by reasoning why this must be true. The given equation is a quadratic equation. The discriminant of this quadratic equation, derived from coefficients $a=1$, $b=2$, and $c=5$, is $\Delta = b^2 - 4ac = (2)^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16$. Since the discriminant is negative, the equation has no real solutions. This justifies that the original statement is indeed false for all real numbers $x$, thereby confirming the correctness of the negated statement.

In conclusion, the negation is: $\boxed{\text{For all } x \in \mathbb{R}, x^{2}+2x+5 \neq 0}$.