\[{x}^{2}-3x-4>0\]
1
Factor x2−3x−4{x}^{2}-3x-4.
(x−4)(x+1)>0(x-4)(x+1)>0

2
Solve for xx.
x=4,−1x=4,-1

3
From the values of xx above, we have these 3 intervals to test.
x<−1−1<x<4x>4\begin{aligned}&x<-1\\&-1<x<4\\&x>4\end{aligned}

4
Pick a test point for each interval.
For the interval x<−1x<-1:

Let's pick x=−2x=-2. Then, (−2)2−3×−2−4>0{(-2)}^{2}-3\times -2-4>0.
After simplifying, we get 6>06>0, which is
true
.
Keep this interval.
.

For the interval −1<x<4-1<x<4:

Let's pick x=0x=0. Then, 02−3×0−4>0{0}^{2}-3\times 0-4>0.
After simplifying, we get −4>0-4>0, which is
false
.
Drop this interval.
.

For the interval x>4x>4:

Let's pick x=5x=5. Then, 52−3×5−4>0{5}^{2}-3\times 5-4>0.
After simplifying, we get 6>06>0, which is
true
.
Keep this interval.
.

5
Therefore,
x<−1,x>4x<-1,x>4

Done
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