we have the system
y < -x^2-x+8 ------> inequality A
the solution for this inequality is the shaded area below the dashed quadratic equation
and
y > x^2+2 -----> inequality B
the solution for this inequality is the shaded area above the dashed quadratic equation
therefore
the solution of the system is the intersection of both areas
see the attached figure to better understand the problem
To find out the intersection points of both graphs, solve the system of equations
y=-x^2-x+8 ------> equation 1
y=x^2+2 ------> equation 2
solve the system by elimination
Adds equation 1 and equation 2
y=-x^2-x+8
y=x^2+2
----------------
2y=-x+10 ------> y=-(1/2)x+5 -----> equation 3
substitute the equation 3 in equation 1 or equation 2 to obtain the values of x
-(1/2)x+5=x^2+2
x^2+(1/2)x-3=0 ------> solve the quadratic equations
the values of x are
x=-2 and x=1.5
to obtain the values of y
substitute the values of x in equation 3
so
For x=-2
y=-(1/2)(-2)+5 ------> y=6
point (-2,6)
For x=1.5
y=-(1/2)(1.5)+5 ------> y=4.25
point (1.5,4.25)
the other two points are the y-intercepts of the quadratic equations