Least squares fitting of a quadratic function through the origin

Ideas
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This project is for educational purposes and can be done more directly by existing functionality.
This post refers more to students trying to figure out how to solve the following problem for the first time

Open in Wolfram Cloud ↗
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Might want to mention built-in capabilities as well as the PseudoInverse function. Or at least note that this is for educational purposes and can be done more directly by existing functionality.

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I’m lazier than you.

data = {{-1, 3}, {1, 1}, {2, 5}};
lm = LinearModelFit[data, {x, x^2}, x, IncludeConstantBasis -> False, 
  WorkingPrecision -> \[Infinity]];
lm["BestFitParameters"]
(* {-(12/11), 20/11} *)
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You are right. I’ll mention it at the beginning of the post. Thank you very much