Question 82567
x(x + 1)(x - 4)
Zeros (x - intercepts)
x = 0
x = -1
x = 4

f(x) > 0
x(x + 1)(x - 4) > 0

X-Axis
<---------(-1)-------(0)-------------------(4)--------------->

Test: x = -2
(-2)(-2 + 1)(-2 - 4) > 0
(-2)(-1)(-6) > 0
-12 > 0 False

Test: x = -0.5
(-0.5)(-0.5 + 1)(-0.5 - 4) > 0
(-0.5)(0.5)(-4.5) > 0
1.125 > 0 True

Test: x = 1
(1)(1 + 1)(1 - 4) > 0
(1)(2)(-3) > 0
-6 > 0 False

Test: x = 5
(5)(5 + 1)(5 - 4) > 0
(5)(6)(1) > 0
30 > 0 True

X-Axis
<---------(-1)========(0)-------------------(4)==============>
(-1,0)U(4,+{{{infinity}}})

Proof:

{{{graph(300,300,-5,5,-5,5,x^3 - 4x^2 + x^2 - 4x)}}}