We investigate the effectiveness of polynomial feature engineering when combined with analytical ridge regression for multi-class classification tasks. Using the NASA Shuttle dataset as a case study, we demonstrate that degree-4 polynomial features enable closed-form solutions to achieve 99.43% test accuracy in 45 milliseconds of training time. This accuracy matches or exceeds previously reported results while offering substantial computational advantages through elimination of iterative optimization. Our systematic evaluation across six feature configurations reveals that test accuracy improves monotonically from 87.33% withlinear features to 99.43% with degree-4 polynomial interactions, representing a 12.10% absolute improvement. Generalization gaps remain below 0.3% across all tested configurations, indicating robust performance despite increased model capacity.These findings suggest that ex-plicit polynomial feature expansion, when properly regularized, provides a computationally efficient alternative to iterative learning methods for problems with polynomial structure. We discuss the applicability of this approach to safety-critical aerospace applications where deterministic training guarantees and rapid model updates are valued.