We need to simplify each of the following expressions and then arrange them in increasing order based on the coefficient of [tex]n^2[/tex]. So, we have:
PART 1
1st expression:
[tex]-5(n^3-n^2-1)+n(n^2-n) \\ \\ -5n^3+5n^2+5+n^3-n^2 \\ \\ \boxed{-4n^3+4n^2+5} \\ \\ \\ Coefficient \ of \ n^2:\mathbf{4}[/tex]
2nd expression:
[tex](n^2-1)(n+2)-n^2(n-3) \\ \\ n^3 + 2n^2-n-2-n^3+3n^2 \\ \\ \boxed{5n^2-n-2} \\ \\ \\ Coefficient \ of \ n^2:\mathbf{5}[/tex]
3rd expression:
[tex]n^2(n-4)+5n^3-6 \\ \\ n^3-4n^2+5n^3-6 \\ \\ \boxed{6n^3-4n^2-6} \\ \\ \\ Coefficient \ of \ n^2:\mathbf{-4}[/tex]
4rd expression:
[tex]2n(n^2-2n-1)+3n^2 \\ \\ 2n^3-4n^2-2n+3n^2 \\ \\ \boxed{2n^3-n^2-2n} \\ \\ \\ Coefficient \ of \ n^2:\mathbf{-1}[/tex]
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PART 2
By arranging these expressions in increasing order based on the coefficient of [tex]n^2[/tex] we have:
[tex](1) \ Coefficient \ \mathbf{-4} \ \rightarrow \boxed{6n^3-4n^2-6} \\ \\ (2) \ Coefficient \ \mathbf{-1} \ \rightarrow \boxed{2n^3-n^2-2n} \\ \\ (3) \ Coefficient \ \mathbf{4} \ \rightarrow \boxed{-4n^3+4n^2+5} \\ \\ (4) \ Coefficient \ \mathbf{5} \ \rightarrow \boxed{5n^2-n-2}[/tex]
