= Solution
Every finite-dimensional $\mathbb C[X]$-<module> is a finitely generated torsion <module> over the <principal ideal domain> $\mathbb C[X]$. The <structure theorem for finitely generated modules over a principal ideal domain> gives
$$
M\cong\bigoplus_{j=1}^r\mathbb C[X]/(p_j),\qquad
p_1\mid p_2\mid\cdots\mid p_r,
$$
where the $p_j$ are uniquely determined monic nonconstant <polynomials>. Equivalently, factoring over $\mathbb C$ and using the <Chinese remainder theorem> gives
$$
\boxed{M\cong\bigoplus_{\lambda\in\mathbb C}\ \bigoplus_j
\mathbb C[X]/((X-\lambda)^{m_{\lambda j}}).}
$$
There are finitely many summands, and the multisets of positive integers $m_{\lambda j}$ are uniquely determined. Each displayed primary cyclic summand is an <indecomposable module>.
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