Find the Area bounded by $y=-x^2+4$, $y=x+2$ I set the two polynomials equal to each other and after multiplying everything by $-1$ I got $x^2+x-2=0$ 
My points of intersection are $x=1$ and $x=-2$
However, the graph of the polynomial is concave up. Why? 

Update: Setting your new equations equal yields $x^2+x-2=0$, which factors as $(x+2)(x-1)$. Now, your intersection points are correct. Also, $-x^2+4>x+2$ from $-2$ to $1$. Now, take
$$\int_{-2}^{1} -x^2-4-(x+2) \mathrm{d}x= \frac{-39}{2}$$
