Answer
Solution set: $ (-2,0)$
Work Step by Step
Follow the "Procedure for Solving Polynomial lnequalities",\ p.412:
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$x^{2}+2x < 0$
$f(x)=x^{2}+2x \quad $...factor the trinomial...
$f(x)=x(x+2)$
$x(x+2) < 0$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$x(x+2) =0$
$x=-2$ or $x=0$
3. Locate these boundary points on a number line, thereby dividing the number line into intervals.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & (-\infty, -2) & (-2,0) & (0,\infty)\\
a=test.val. & -3 & -1 & 1\\
f(a) & (-3)(-1) & (-1)(-1) & (1)(3)\\
f(a) < 0 ? & F & T & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $ (-2,0)$