${x}^{2}+8x+15 > 0$
Write $8x$ as a sum${x}^{2}+5x+3x+15 > 0$
Factor out $x$ from the expression$x \times \left( x+5 \right)+3x+15 > 0$
Factor out $3$ from the expression$x \times \left( x+5 \right)+3\left( x+5 \right) > 0$
Factor out $x+5$ from the expression$\left( x+5 \right) \times \left( x+3 \right) > 0$
Separate the inequality into two possible cases$\begin{array} { l }\left\{\begin{array} { l } x+5 > 0 \\ x+3 > 0\end{array} \right.,\\\left\{\begin{array} { l } x+5 < 0 \\ x+3 < 0\end{array} \right.\end{array}$
Solve the inequality for $x$$\begin{array} { l }\left\{\begin{array} { l } x > -5 \\ x+3 > 0\end{array} \right.,\\\left\{\begin{array} { l } x+5 < 0 \\ x+3 < 0\end{array} \right.\end{array}$
Solve the inequality for $x$$\begin{array} { l }\left\{\begin{array} { l } x > -5 \\ x > -3\end{array} \right.,\\\left\{\begin{array} { l } x+5 < 0 \\ x+3 < 0\end{array} \right.\end{array}$
Solve the inequality for $x$$\begin{array} { l }\left\{\begin{array} { l } x > -5 \\ x > -3\end{array} \right.,\\\left\{\begin{array} { l } x < -5 \\ x+3 < 0\end{array} \right.\end{array}$
Solve the inequality for $x$$\begin{array} { l }\left\{\begin{array} { l } x > -5 \\ x > -3\end{array} \right.,\\\left\{\begin{array} { l } x < -5 \\ x < -3\end{array} \right.\end{array}$
Find the intersection$\begin{array} { l }x \in \langle-3, +\infty\rangle,\\\left\{\begin{array} { l } x < -5 \\ x < -3\end{array} \right.\end{array}$
Find the intersection$\begin{array} { l }x \in \langle-3, +\infty\rangle,\\x \in \langle-\infty, -5\rangle\end{array}$
Find the union$x \in \langle-\infty, -5\rangle \cup \langle-3, +\infty\rangle$