Reconciling OLS as a linear regression model and polynomial regressors A bit of a naive question.
I understand that OLS is used for a linear regression model (for example, Wikipedia page for OLS:  OLS is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares).
I, therefore, don't understand why OLS can be used to estimate what I understand to be non-linear regressions, s.t. $$y= a+bx+bx^2.$$
What am I missing?

You are missing the definition of linear regression. By linear regression, it is meant any regression equation of the form
$$
y = \beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_px_p + \epsilon,\tag{*}
$$
where $\beta_i$ are the regression parameters, $x_j$ are the covariates and $\epsilon$ is an error term. The keyword linear has to do with how each $\beta_i$ must enter in (*), i.e. they must enter linearly.
For example
$$
y = \beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_1x_2+\epsilon,
$$
is a linear regression model, whereas the equation
$$
y = \beta_0+x_1^{\beta_1}+\beta_2x_2+\cdots+\beta_px_p + \epsilon, \tag{**}
$$
does not identify a linear regression model. We can still use OLS for (**), though the solution won't necessarily have a closed-form expression as in (*).
