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Simplify each expression, and then arrange them in increasing order based on the coefficient of [tex]n^2[/tex].

[tex]\[
\begin{array}{c}
-5\left(n^3 - n^2 - 1\right) + n\left(n^2 - n\right) \\
\left(n^2 - 1\right)(n + 2) - n^2(n - 3) \\
n^2(n - 4) + 5n^3 - 6 \\
2n\left(n^2 - 2n - 1\right) + 3n^2
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's simplify each expression and arrange them in increasing order based on the coefficient of [tex]\( n^2 \)[/tex].

Given expressions:
1. [tex]\(-5(n^3 - n^2 - 1) + n(n^2 - n)\)[/tex]
2. [tex]\((n^2 - 1)(n + 2) - n^2(n - 3)\)[/tex]
3. [tex]\(n^2(n - 4) + 5n^3 - 6\)[/tex]
4. [tex]\(2n(n^2 - 2n - 1) + 3n^2\)[/tex]

### Step 1: Simplify each expression

1. Simplify [tex]\(-5(n^3 - n^2 - 1) + n(n^2 - n)\)[/tex]:
[tex]\[ -5(n^3 - n^2 - 1) + n(n^2 - n) = -5n^3 + 5n^2 + 5 + n^3 - n^2 = -4n^3 + 4n^2 + 5 \][/tex]

2. Simplify [tex]\((n^2 - 1)(n + 2) - n^2(n - 3)\)[/tex]:
[tex]\[ (n^2 - 1)(n + 2) - n^2(n - 3) = (n^3 + 2n^2 - n - 2) - (n^3 - 3n^2) = 5n^2 - n - 2 \][/tex]

3. Simplify [tex]\(n^2(n - 4) + 5n^3 - 6\)[/tex]:
[tex]\[ n^2(n - 4) + 5n^3 - 6 = n^3 - 4n^2 + 5n^3 - 6 = 6n^3 - 4n^2 - 6 \][/tex]

4. Simplify [tex]\(2n(n^2 - 2n - 1) + 3n^2\)[/tex]:
[tex]\[ 2n(n^2 - 2n - 1) + 3n^2 = 2n^3 - 4n^2 - 2n + 3n^2 = 2n^3 - n^2 - 2n \][/tex]

### Step 2: Extract the coefficient of [tex]\( n^2 \)[/tex]

1. [tex]\[-4n^3 + 4n^2 + 5: \quad \text{Coefficient of } n^2 = 4\][/tex]
2. [tex]\[5n^2 - n - 2: \quad \text{Coefficient of } n^2 = 5\][/tex]
3. [tex]\[6n^3 - 4n^2 - 6: \quad \text{Coefficient of } n^2 = -4\][/tex]
4. [tex]\[2n^3 - n^2 - 2n: \quad \text{Coefficient of } n^2 = -1\][/tex]

### Step 3: Arrange in increasing order based on the coefficient of [tex]\( n^2 \)[/tex]

- Coefficient of [tex]\(-4n^2\)[/tex]: [tex]\(6n^3 - 4n^2 - 6\)[/tex]
- Coefficient of [tex]\(-n^2\)[/tex]: [tex]\(2n^3 - n^2 - 2n\)[/tex]
- Coefficient of [tex]\(4n^2\)[/tex]: [tex]\(-4n^3 + 4n^2 + 5\)[/tex]
- Coefficient of [tex]\(5n^2\)[/tex]: [tex]\(5n^2 - n - 2\)[/tex]

### Final Sorted List:

1. [tex]\(6n^3 - 4n^2 - 6\)[/tex]
2. [tex]\(2n^3 - n^2 - 2n\)[/tex]
3. [tex]\(-4n^3 + 4n^2 + 5\)[/tex]
4. [tex]\(5n^2 - n - 2\)[/tex]

Arranged in increasing order based on the coefficient of [tex]\( n^2 \)[/tex], the expressions are:

1. [tex]\(n^2(n - 4) + 5n^3 - 6\)[/tex]
2. [tex]\(2n(n^2 - 2n - 1) + 3n^2\)[/tex]
3. [tex]\(-5(n^3 - n^2 - 1) + n(n^2 - n)\)[/tex]
4. [tex]\((n^2 - 1)(n + 2) - n^2(n - 3)\)[/tex]

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