423ffd…: “Like a great house”

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Suppose $T:\mathbb {P}^2\rightarrow \mathbb {P}^3$ is a linear transformation such that

\[T(x^2)=x^3,\quad T(x+1)=0,\quad T(x-1)=x\]

Find $T(x^2+x+1)$.

\[T(x^2+x+1)=\answer {x^3}\]

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 7.1.4 (c).