Answer
$(-\infty, -2) \cup \left(-1, 1\right) \cup(1, \infty) $
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 Rational function.
$\displaystyle \frac{(x-1)(x-1)}{(x+1)(x+2)}\gt 0$,
$f(x)=\displaystyle \frac{(x-1)(x-1)}{(x+1)(x+2)}\gt 0$,
2.The cut points are:
$\displaystyle \frac{(x-1)(x-1)}{(x+1)(x+2)}=0$
$x=-2$ or or $x=-1$ or $x=1$
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,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \gt 0 ? \\
& & \displaystyle \displaystyle \frac{(a-1)(a-1)}{(a+1)(a+2)}& \\
(-\infty,-2) & -5 & \frac{(-)(-)}{(-)(-)}=(+) & T\\
(-2, -1) & -\frac{3}{2} & \frac{(-)(-)}{(-)(+)}=(-) & F\\
(-1, 1) & 0 & \frac{(-)(-)}{(+)(+)}=(+) & T\\
(1, \infty) & 5 & \frac{(+)(+)}{(+)(+)}=(+) & T\\
\end{array}$
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: $(-\infty, -2) \cup \left(-1, 1\right) \cup(1, \infty) $
