Solve for

|x(x+1)| < |x +4|

[x(x+1)] 2 < (x+4)2

[x(x+1)] 2 – (x+4)2 < 0

[x (x+1) – (x+4)] [x (x+1) + (x+4)] < 0

(x2 + x – x – 4) (x2 + x + x + 4) < 0

(x2 – 4) (x2 + 2x + 4) < 0

the product of two factors is negative since it is less than (<) zero (0).

and x2 + 2x + 4 = x2 + 2x + 1 + 3

= (x + 1)2 + 3 then it is positive.

then (x2 – 4) is negative.

Finally:

x2 – 4 < 0

x2 < 4

take the sq. root on both sides, then

|x| < 2 Q.E.D