Step 1: Understand the Reaction.
LiAlH\(_4\) reacts with Me\(_3\)N·HCl to form a tetrahedral complex. Upon further reaction with another equivalent of Me\(_3\)N·HCl, a new compound is formed, which contains two NMe\(_3\) groups. The geometry of this compound is trigonal bipyramidal.
Step 2: Analyze the Geometry.
The presence of two NMe\(_3\) groups coordinated to the Al center results in a trigonal bipyramidal geometry, as the two bulky NMe\(_3\) groups need to be placed in axial positions to minimize steric repulsion.
Step 3: Conclusion.
The correct answer is \( \text{AlH}_3(\text{NMe}_3)_2 \) with trigonal bipyramidal geometry.
Final Answer: \[ \boxed{\text{AlH}_3(\text{NMe}_3)_2 \text{ and trigonal bipyramidal}} \]