Polynomial features: expanding inputs before a linear model

The problem, from first principles

linear (this curriculum's very first question) can only ever compute weighted SUMS of its inputs, it fundamentally cannot represent a curve, no matter how it's trained. Feature engineering: deriving a feature that makes the model's job easier (Math & Statistics) already showed one fix: hand-derive a specific nonlinear feature (radius_feature) when you know exactly what shape the data needs. Polynomial feature expansion generalizes that idea into something systematic, instead of hand-picking one clever feature, generate EVERY power and cross-product of your existing features up to some degree, and let a plain linear model choose which of those (now numerous) features actually matter.

07-evaluation/03-bias-variance-tradeoff's own polynomial_features already did this for a SINGLE input variable. This question generalizes it to MULTIPLE features at once, which introduces a genuinely new kind of term that a single-variable expansion never needs: INTERACTION terms, products of two DIFFERENT features (x0 * x1), not just powers of one feature alone (x0^2).

From theory to code

Theory enumerates every monomial (a product of input features raised to some powers) of total degree 0 through degree, using itertools.combinations_with_replacement over feature indices, the systematic way to generate "every way to multiply some number of features together, repeats allowed, order doesn't matter."

Implement polynomial_features_multivariate(X, degree) against that reasoning. The signature and docstring are already in the editor.

Constraints

• X is (n, num_features), any number of features, not just one.
• Output columns must be ordered by degree first (0, then 1, then 2, ...), then by combinations_with_replacement order within each degree, matching sklearn.preprocessing.PolynomialFeatures' own column ordering exactly.
• Degree 0 always produces exactly one column, all 1s (the bias/intercept term).

Hints

Open one at a time. Each gives away a little more than the last.