This solution will guide you through the process of solving the given problem by systematically applying the standard formulas for hyperbolas and ellipses. We will focus on clarity, detailed explanations, and proper mathematical notation.

1. Understanding the Problem and Key Concepts

The problem asks us to calculate the value of $12(e_H^2 + e_E^2)$, where $e_H$ and $e_E$ are the eccentricities of a given hyperbola $H$ and an ellipse $E$, respectively. A crucial piece of information is that the length of the latus rectum of $H$ is equal to the length of the latus rectum of $E$.

To solve this, we need to recall the standard forms and associated formulas for hyperbolas and ellipses, specifically