$$f(x)=a_3x^3+a_2x^2+2x+8$$
$$f(2)=0=8a_3+4a_2+4+8=8a_3+4a_2+12\Rightarrow 0=2a_3+a_2+3$$
$$f(-1)=12=-a_3+a_2-2+8=-a_3+a_2+6\Rightarrow 0-a_3+a_2-6 $$
Beide Gleichungen subtrahieren:
$$ 0=~2a_3+a_2+3\\ 0=-a_3+a_2-6$$
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$$0=3a_3+9\Rightarrow a_3=-3$$
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$$a_2=3$$
$$f(x)=-3x^3+3x^2+2x+8$$