**First Route:**

x^2 -4x +8 < x(x+8)

x^2 -4x +8 < x^2 +8x

Now the trouble starts:

x^2 -x^2 -4x +4x +8 < x^2 -x^2 +8x +4x

8 < 12x

8/12 < 12/12 x

(2/3) < x

...0, ((2/3), 1, 2, 3, ...+inf.

((2/3),+inf.)

**Second route:**

x^2 -4x +8 < x(x+8)

x^2 -4x +8 < x^2 +8x

Alternate:

x^2 -x^2 -4x -8x +8 -8 < x^2 -x^2 +8x -8x -8

4x < -8

x < -2

-inf.... -4, -3, -2,) -1, 0, ...

(-inf.,-2)

Now I know having 4x be less than -8 would in most cases result in an incorrect assessment of the equation. However, this situation begs the question, "Is there a convention for determining what gets moved in what order?".