Key Concepts and Properties Used:

This problem involves evaluating a $3 \times 3$ determinant and then extracting coefficients from the resulting polynomial. The key concepts and properties we will utilize are:

1. Determinant Properties:
   • Row/Column Operations: The value of a determinant remains unchanged if we apply the operation $R_i \to R_i + kR_j$ (or $C_i \to C_i + kC_j$). This is crucial for simplifying the determinant by creating zeros or common factors.
   • Factoring out Common Elements: If all elements of a row or column are multiplied by a constant $k$, then the value of the determinant is multiplied by $k$. Conversely, a common factor from any row or column can be taken out of the determinant.
   • Expansion of a Determinant: A $3 \times 3$ determinant can be expanded along any row or column. The expansion along a row $i$ is given by $\sum_{j=1}^{3} a_{ij} C_{ij}$, where $C_{ij} = (-1)^{i+j} M_{ij}$ is the cofactor of the element $a_{ij}$, and $M_{ij}$ is the minor (determinant of the submatrix obtained by deleting row $i$ and column $j$). Expanding along a row/column with more zeros simplifies the calculation significantly.

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