# One Theorem Kills It All

**Algebra**Level 5

Let \(f(x)=x^3+6x^2+3x+7\) and let \(g(x)=x^2+4x+1\). The roots of \(f(x)\) and \(g(x)\) are plotted on the complex plane. A triangle is drawn with vertices at the roots of \(f(x)\) and an ellipse is drawn with foci at the roots of \(g(x)\) that is tangent to the triangle at the midpoint of the side that connects the two complex roots. Find the number of intersection points of the triangle and ellipse.

**Details and Assumptions**

- The tangency point counts as an intersection point.