= Solution
A natural next fit is <quadratic regression>, which remains a <normal linear model> in its unknown <regression coefficients>:
$$
\boxed{Y_i=\beta_0+\beta_1x_i+\beta_2x_i^2+\varepsilon_i,
\qquad \varepsilon_i\overset{\mathrm{iid}}\sim N(0,\sigma^2).}
$$
The new <design matrix> has rows $(1,x_i,x_i^2)$ and must have <matrix rank> three; three distinct predictor values suffice. The curved <regression residual> pattern suggests trying a positive quadratic term, but its sign and adequacy should be checked after fitting. Inspect the new <regression residuals> to see whether the systematic curvature has disappeared.
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