145184…: “As an exciting cube”

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Find the coordinates of $\vec {v}=ax^2+bx+c$ of $\mathbb {P}^2$ with respect to the ordered basis $\mathcal {B}=\left \{x^2, x+1, x+2\right \}$.

\[[\vec {v}]_{\mathcal {B}}=\begin{bmatrix}\answer {a}\\\answer {2b-c}\\\answer {c-b}\end{bmatrix}\]

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 9.1.1 (b).