Back in grad school when we covered basic OLS methods expanding the feature space using a quadratic manifold was a common technique for teaching that regression can support nonlinear features and still remain "linear in the coefficients".

It is also implemented in widely used ML libraries. Scikit-learn, for example, has a KernelPCA object which supports using a polynomial kernel (in this instance a degree two) that computes the inner product on a explicit feature map which contains all second-order monomials.