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: $-\left(x+4\right)\left(x-3\right) > 0$. The minus multiplies the product by -1, so for the whole thing to be positive, the product $\left(x+4\right)\left(x-3\right)$ 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: $\left(x+4\right)\left(x-3\right) < 0$.