X^3+x^2<4x+4 solve the inequality algebraically?

First, get out of the habit of hitting Shift just because you start a new line. X is not x and a math program will not tell you that you misspelled something, it will just give you goofy results. You can not stand spelling mistrakes in math.

Change the inequality to equality and solve. Then consider the result in the original statement.

x^3 + x^2 = 4x + 4

x^3 + x^2 - 4x - 4 = 0

(x + 2)(x - 2)(x + 1) = 0

Looks like this:
www.wolframalpha.com/input/?i=solve+x%5E3+%2B+x%5E2+-+4x+-+4+%3D+0

That tells you the points of interest when you plot the inequality:
www.wolframalpha.com/input/?i=plot+x%5E3%2Bx%5E2%3C4x%2B4

Here is what you get the way you first typed the problem with capital X:
www.wolframalpha.com/input/?i=plot+X%5E3%2Bx%5E2%3C4x%2B4