Conic Complex

Geometry Level 4

$\begin{aligned} f(x , y) &= nx^2 + \left(\dfrac{nm}{2}\right)xy + my^2 + 4x + 2y + 1 = 0\\ g(x , y) &= nx^2 + (n + m)xy + my^2 + 4x + 2y + 1 = 0\\ h(x , y) &= nx^2 + (n - m)xy + my^2 + 4x + 2y + 1 = 0 \end{aligned}$

Let the graphs $f, g, h$ be a parabola, a hyperbola, and an ellipse, respectively.

If $n$ and $m$ are positive integers satisfying the constraints above, compute $2n + m$ .