Answer
$x=-\displaystyle \frac{5}{2}+\frac{\sqrt{13}}{2}$ or $x=-\displaystyle \frac{5}{2}-\frac{\sqrt{13}}{2}$
Work Step by Step
$ x^{2}+5x+3=0\qquad$... solve with the Quadractic formula. $a=1,\ b=5,\ c=3$
$ x=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\qquad$... substitute $b$ for $5,\ a$ for $1$ and $c$ for $3$.
$ x=\displaystyle \frac{-5\pm\sqrt{(5)^{2}-4\cdot(3)\cdot 1}}{2\cdot 1}\qquad$... simplify.
$x=\displaystyle \frac{-5\pm\sqrt{25-12}}{2}$
$ x=\displaystyle \frac{-5\pm\sqrt{13}}{2}\qquad$... the symbol $\pm$ indicates two solutions.
$x=-\displaystyle \frac{5}{2}+\frac{\sqrt{13}}{2}$ or $x=-\displaystyle \frac{5}{2}-\frac{\sqrt{13}}{2}$