Answer
$\left(-\sqrt 2, \sqrt 2\right) $
Work Step by Step
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.
$x^2e^x-2e^x<0,$
$e^x(x^2-2) < 0,$
$e^x(x-\sqrt 2)(x+\sqrt 2) < 0,$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$e^x(x-\sqrt 2)(x+\sqrt 2)<0,$
$x=-\sqrt 2$ or $x=\sqrt 2$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& &(e^a)(a-\sqrt 2)(a+\sqrt 2)& \\
(-\infty, -\sqrt 2) & -5 & (+)(-)(-)=(+) & F\\
(-\sqrt 2, \sqrt 2) & 0 & (+)(-)(+)=(-) & T\\
(\sqrt 2,\infty) & 5 & (+)(+)(+)=(+) & F
\end{array}$
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: $\left(-\sqrt 2, \sqrt 2\right) $
