jmc

The points of intersection of the line $y=7$ and $y=2x^2+8x+4$, by substitution, are found when $2x^2+8x+4=7⟹2x^2+8x-3=0$. By the quadratic formula, $$x=\frac{-8\pm \sqrt{8^2-4\cdot 2\cdot (-3)}}{2\cdot 2}.$$We want to find the difference of these roots to find the difference of the x-coordinates of the point of intersection, which will give a side length of the square. The difference is given by $\frac{\sqrt{8^2-4\cdot 2\cdot (-3)}}{2}=\frac{\sqrt{88}}{2}=\sqrt{22}$. Thus, the area of the square is $\boxed{22}$.