<s>[INST] How do I find $f(x)$ when provided $g(x)$ and $(f\circ g)(x)$? Let $g(x) = 2x^2−2x−1$. 
Find a function $f$ such that $(f\circ g)(x) = −8x^6 + 24x^5 −8x^4 − 24x^3 + 6x^2 + 10x + 6.$
$f(x) =$
 [/INST] Let's suppose that $f(x)=Ax^3+Bx^2+Cx+D$.  
Because $2^3=8$, we can see that $A=-1$.  
So let's substract $-(2x^2-2x-1)^3=-8x^6+24x^5-12x^4-16x^3+6x^2+6x+1$ from $−8x^6 + 24x^5 −8x^4 − 24x^3 + 6x^2 + 10x + 6$.  The result is $4x^4-8x^3+4x+5$.  
$2^2=4$, so $B=1$. Now substract $(2x^2-2x-1)^2$ from $4x^4-8x^3+4x+5$. The result is $4$. So $C=0$ and $D=4$.
So the function you are looking for is: $$f(x)=-x^3+x^2+4$$
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