Positive times negative is negative so we don't want that case. Positive times positive is positive and negative times negative is positive. But [tex]x^2+ 4[/tex] is never negative so x must be non-negative.
Another way of thinking about this is that dividing both sides of an inequality by a positive number does not change the direction of the inequality. So dividing both sides of [tex]x(x^2+ 4)\ge 0[/tex] by the positive number [tex]x^2+ 4[/tex] gives [tex]x\ge 0[/tex].