Let's simplify each expression one by one and determine their coefficients for [tex]\( n^2 \)[/tex]. Finally, we will arrange the simplified expressions in increasing order based on the coefficients of [tex]\( n^2 \)[/tex].
1. Simplify:
[tex]\[
-5(n^3 - n^2 - 1) + n(n^2 - n)
\][/tex]
[tex]\[
= -5n^3 + 5n^2 + 5 + n^3 - n^2
\][/tex]
[tex]\[
= -4n^3 + 4n^2 + 5
\][/tex]
The coefficient of [tex]\( n^2 \)[/tex] is 4.
2. Simplify:
[tex]\[
(n^2 - 1)(n + 2) - n^2(n - 3)
\][/tex]
[tex]\[
= n^3 + 2n^2 - n - 2 - n^3 + 3n^2
\][/tex]
[tex]\[
= 5n^2 - n - 2
\][/tex]
The coefficient of [tex]\( n^2 \)[/tex] is 5.
3. Simplify:
[tex]\[
n^2(n - 4) + 5n^3 - 6
\][/tex]
[tex]\[
= n^3 - 4n^2 + 5n^3 - 6
\][/tex]
[tex]\[
= 6n^3 - 4n^2 - 6
\][/tex]
The coefficient of [tex]\( n^2 \)[/tex] is -4.
4. Simplify:
[tex]\[
2n(n^2 - 2n - 1) + 3n^2
\][/tex]
[tex]\[
= 2n^3 - 4n^2 - 2n + 3n^2
\][/tex]
[tex]\[
= 2n^3 - n^2 - 2n
\][/tex]
The coefficient of [tex]\( n^2 \)[/tex] is -1.
Now, let's arrange the simplified expressions in increasing order based on the coefficient of [tex]\( n^2 \)[/tex]:
1. [tex]\( 6n^3 - 4n^2 - 6 \)[/tex] [tex]\(\quad (\text{coefficient of } n^2 \text{ is } -4)\)[/tex]
2. [tex]\( 2n^3 - n^2 - 2n \)[/tex] [tex]\(\quad (\text{coefficient of } n^2 \text{ is } -1)\)[/tex]
3. [tex]\( -4n^3 + 4n^2 + 5 \)[/tex] [tex]\(\quad (\text{coefficient of } n^2 \text{ is } 4)\)[/tex]
4. [tex]\( 5n^2 - n - 2 \)[/tex] [tex]\(\quad (\text{coefficient of } n^2 \text{ is } 5)\)[/tex]
Thus, the expressions in increasing order based on the coefficient of [tex]\( n^2 \)[/tex] are:
[tex]\[
6n^3 - 4n^2 - 6, \quad 2n^3 - n^2 - 2n, \quad -4n^3 + 4n^2 + 5, \quad 5n^2 - n - 2
\][/tex]
