Alright, let's simplify each expression step by step and then arrange them based on the coefficient of [tex]\(n^2\)[/tex].
1. Simplify the expression [tex]\(-5(n^3-n^2-1)+n(n^2-n)\)[/tex]:
[tex]\[
\begin{aligned}
&-5(n^3 - n^2 - 1) + n(n^2 - n) \\
&= -5n^3 + 5n^2 + 5 + n^3 - n^2 \\
&= -4n^3 + 4n^2 + 5.
\end{aligned}
\][/tex]
2. Simplify the expression [tex]\((n^2-1)(n+2)-n^2(n-3)\)[/tex]:
[tex]\[
\begin{aligned}
&(n^2 - 1)(n + 2) - n^2(n - 3) \\
&= (n^3 + 2n^2 - n - 2) - (n^3 - 3n^2) \\
&= n^3 + 2n^2 - n - 2 - n^3 + 3n^2 \\
&= 5n^2 - n - 2.
\end{aligned}
\][/tex]
3. Simplify the expression [tex]\(n^2(n-4) + 5n^3 - 6\)[/tex]:
[tex]\[
\begin{aligned}
&n^2(n - 4) + 5n^3 - 6 \\
&= n^3 - 4n^2 + 5n^3 - 6 \\
&= 6n^3 - 4n^2 - 6.
\end{aligned}
\][/tex]
4. Simplify the expression [tex]\(2n(n^2-2n-1)+3n^2\)[/tex]:
[tex]\[
\begin{aligned}
&2n(n^2 - 2n - 1) + 3n^2 \\
&= 2n^3 - 4n^2 - 2n + 3n^2 \\
&= 2n^3 - n^2 - 2n.
\end{aligned}
\][/tex]
Now let's arrange these simplified expressions in increasing order based on the coefficient of [tex]\(n^2\)[/tex].
- From [tex]\(-4n^3 + 4n^2 + 5\)[/tex], the coefficient of [tex]\(n^2\)[/tex] is [tex]\(4\)[/tex].
- From [tex]\(5n^2 - n - 2\)[/tex], the coefficient of [tex]\(n^2\)[/tex] is [tex]\(5\)[/tex].
- From [tex]\(6n^3 - 4n^2 - 6\)[/tex], the coefficient of [tex]\(n^2\)[/tex] is [tex]\(-4\)[/tex].
- From [tex]\(2n^3 - n^2 - 2n\)[/tex], the coefficient of [tex]\(n^2\)[/tex] is [tex]\(-1\)[/tex].
Arranging them in increasing order based on the coefficient of [tex]\(n^2\)[/tex]:
1. [tex]\(6n^3 - 4n^2 - 6\)[/tex] (coefficient [tex]\(-4\)[/tex])
2. [tex]\(2n^3 - n^2 - 2n\)[/tex] (coefficient [tex]\(-1\)[/tex])
3. [tex]\(-4n^3 + 4n^2 + 5\)[/tex] (coefficient [tex]\(4\)[/tex])
4. [tex]\(5n^2 - n - 2\)[/tex] (coefficient [tex]\(5\)[/tex])
So the final order of the expressions simplified and arranged is:
[tex]\[
6n^3 - 4n^2 - 6, \quad 2n^3 - n^2 - 2n, \quad -4n^3 + 4n^2 + 5, \quad 5n^2 - n - 2.
\][/tex]
