What is linear, exactly?

If we label a model like y = a + bx as linear, no eyebrow should be raised. Now, consider a regression model involving a squared explanatory variable:

Is this linear? Actually it is, in terms of the factor that matters most: fitting model coefficients. True, the model is nonlinear in terms of the explanatory variable x, but the actual unknowns when we apply least squares are the coefficients a, b₁, and b₂. In this respect, the model is linear and we may easily apply the multivariate least-squares approach, which we will illustrate. If we want to tell the difference between the two models above, we may say that the first one is linear and first-order, whereas the second one is linear and second-order. In a polynomial model like (10.6), the order of the model is the largest degree occurring in the polynomial.