A Negative Leading Term

In -x^2 - x + 12 > 0 the squared term has a minus in front. Factoring keeps that minus outside the brackets: -(x + 4)(x - 3) > 0. The minus multiplies the product by -1, so for the whole thing to be positive, the product (x + 4)(x - 3) must be negative.

So the picture flips. Without the minus, this product is positive outside the roots; with it, the expression is positive between them. The factors need opposite signs: x + 4 > 0 and x - 3 < 0 gives -4 < x < 3, and the other case, x below -4 and above 3 at once, is empty.

The graph agrees. A negative squared term makes the parabola open down, so it sits above the x-axis only between its roots. Multiplying both sides by -1 first works too, as long as you flip the sign: (x + 4)(x - 3) < 0.