Sure, let’s solve and simplify each expression step-by-step, and then arrange them based on the coefficient of [tex]\(n^2\)[/tex].
### Expression 1
[tex]\[
-5(n^3 - n^2 - 1) + n(n^2 - n)
\][/tex]
First, expand the terms:
[tex]\[
-5n^3 + 5n^2 + 5 + n^3 - n^2
\][/tex]
Combine like terms:
[tex]\[
(-5n^3 + n^3) + (5n^2 - n^2) + 5 = -4n^3 + 4n^2 + 5
\][/tex]
### Expression 2
[tex]\[
(n^2 - 1)(n + 2) - n^2(n - 3)
\][/tex]
Expand each part:
[tex]\[
(n^2 \cdot n + n^2 \cdot 2 - 1 \cdot n - 1 \cdot 2) - (n^2 \cdot n - n^2 \cdot 3)
\][/tex]
[tex]\[
(n^3 + 2n^2 - n - 2) - (n^3 - 3n^2)
\][/tex]
Combine the terms:
[tex]\[
(n^3 + 2n^2 - n - 2 - n^3 + 3n^2) = 5n^2 - n - 2
\][/tex]
### Expression 3
[tex]\[
n^2(n - 4) + 5n^3 - 6
\][/tex]
Expand and combine like terms:
[tex]\[
n^3 - 4n^2 + 5n^3 - 6
\][/tex]
Combine like terms:
[tex]\[
(1n^3 + 5n^3) - 4n^2 - 6 = 6n^3 - 4n^2 - 6
\][/tex]
### Expression 4
[tex]\[
2n(n^2 - 2n - 1) + 3n^2
\][/tex]
Expand and combine like terms:
[tex]\[
2n^3 - 4n^2 - 2n + 3n^2
\][/tex]
Combine like terms:
[tex]\[
2n^3 - (4n^2 - 3n^2) - 2n = 2n^3 - n^2 - 2n
\][/tex]
### Extracting Coefficients of [tex]\(n^2\)[/tex]:
1. [tex]\( -4n^3 + 4n^2 + 5 \)[/tex] : Coefficient of [tex]\(n^2\)[/tex] is [tex]\(4\)[/tex]
2. [tex]\( 5n^2 - n - 2 \)[/tex] : Coefficient of [tex]\(n^2\)[/tex] is [tex]\(5\)[/tex]
3. [tex]\( 6n^3 - 4n^2 - 6 \)[/tex] : Coefficient of [tex]\(n^2\)[/tex] is [tex]\(-4\)[/tex]
4. [tex]\( 2n^3 - n^2 - 2n \)[/tex] : Coefficient of [tex]\(n^2\)[/tex] is [tex]\(-1\)[/tex]
### Arranging Based on Coefficients of [tex]\(n^2\)[/tex] (Increasing Order):
1. [tex]\( 6n^3 - 4n^2 - 6 \)[/tex] : Coefficient of [tex]\(n^2 = -4\)[/tex]
2. [tex]\( 2n^3 - n^2 - 2n \)[/tex] : Coefficient of [tex]\(n^2 = -1\)[/tex]
3. [tex]\( -4n^3 + 4n^2 + 5 \)[/tex] : Coefficient of [tex]\(n^2 = 4\)[/tex]
4. [tex]\( 5n^2 - n - 2 \)[/tex] : Coefficient of [tex]\(n^2 = 5\)[/tex]
### Final Answer:
Arrange these expressions from the lowest to the highest coefficient of [tex]\(n^2\)[/tex]:
[tex]\[
\begin{array}{ll}
1. & 6n^3 - 4n^2 - 6\\
2. & 2n^3 - n^2 - 2n\\
3. & -4n^3 + 4n^2 + 5\\
4. & 5n^2 - n - 2
\end{array}
\][/tex]
