# Calculate: x^2+8x >-15

## Expression: ${x}^{2}+8x > -15$

Move the constant to the left-hand side and change its sign

${x}^{2}+8x+15 > 0$

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

$x \in \langle-\infty, -5\rangle \cup \langle-3, +\infty\rangle$

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