To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$x^{2}+2x+3=0$$
All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $1$ for $a$, $2$ for $b$, and $3$ for $c$ in the quadratic formula.
$$x=\frac{-2±\sqrt{2^{2}-4\times 1\times 3}}{2}$$
Do the calculations.
$$x=\frac{-2±\sqrt{-8}}{2}$$
Since the square root of a negative number is not defined in the real field, there are no solutions. Expression $x^{2}+2x+3$ has the same sign for any $x$. To determine the sign, calculate the value of the expression for $x=0$.
$$0^{2}+2\times 0+3=3$$
The value of the expression $x^{2}+2x+3$ is always positive. Inequality holds for $x\in \mathrm{R}$.
