Solving polynomial inequalities requires that we divide the polynomial into discrete intervals.
For example, if we have \( x^2 +6x - 16 > 0\), we can first find the places where the expression is equal to 0. Factoring, we can see that \( (x+8)(x-2) = x^2 +6x -16 \), so the polynomial is equal to 0 at \( x = -8 \) and at \( x = 2 \). This means that on the intervals \( (-\infty, -8), (-8,2),\) and \( (2, \infty) \), the expression will either be only \( >0 \) or \( < 0 \) (because it can't pass through zero on those intervals). Testing each interval, we find that the expression is positive on the outer intervals and is negative on \( (-8,2) \).
Thus the solution is \( x < -8 \) and \( x > 2 \).