Hi Tina,

There's two ways to understand the final answer here.

I would recommend a hybrid algebraic and graphical approach. Factoring like you did, you get that the graph of $y=x^2+x-6$ has zeros at $x=-3$ and $x=2$. Since the graph has a positive $x^2$ coefficient, it opens upward, so the graph looks like this (click on the graph below to view on Desmos):

This is where the "between" comes from. From the graph we can see it will be negative for $x$ values with $-3 < x < 2$. This gives the $4$ values $x=-2, -1, 0, 1$.

For the purely algebraic approach, factoring gives $(x+3)(x-2) < 0$. Here, for the product of two numbers to be negative, we need one to be positive and the other to be negative. Hence,

• $x+3 < 0$ and $x-2 > 0$, implying that $x < -3$ and $x > 2$, but this is impossible.
• $x+3 > 0$ and $x - 2 < 0$, implying $x > -3$ and $x < 2$, so $-3 < x < 2$ as before.

For the purely algebraic approach, you do need to be careful with the "and", as we did in the two cases above. Hope this helps!