Understanding logic of proof

Prove: If $\displaystyle x^2 + x - 6 \geq 0$, then $\displaystyle x \leq -3$ or $\displaystyle x \geq 2.$

Im not understanding the logic behind this questions answer. My professor went ahead and:

Suppose that $\displaystyle x^2 + x - 6 \geq 0 $ and $\displaystyle x > -3$. **It follows that $\displaystyle (x-2)(x+3) \geq 0$ and, since $\displaystyle x + 3 > 0$, it must be that $\displaystyle x -2 \geq 0$. That is, $\displaystyle x \geq 2$.**

im not understanding why they suppose x > -3, and how the bolded makes sense. I think im just not getting something easy here, please help.

Thanks!