Alright, let's go through the process of subtracting the polynomial [tex]\(6x^3 - 2x + 3\)[/tex] from [tex]\(-3x^3 + 5x^2 + 4x - 7\)[/tex]. We will identify and clarify each step Lorne used and arrive at the correct answer.
### Problem Statement:
We need to subtract the polynomial [tex]\(6x^3 - 2x + 3\)[/tex] from [tex]\(-3x^3 + 5x^2 + 4x - 7\)[/tex].
### Step-by-Step Solution:
1. Rewrite each polynomial:
- The first polynomial is [tex]\(-3x^3 + 5x^2 + 4x - 7\)[/tex].
- The second polynomial is [tex]\(6x^3 - 2x + 3\)[/tex].
2. Express the subtraction operation:
We need to perform [tex]\((-3x^3 + 5x^2 + 4x - 7) - (6x^3 - 2x + 3)\)[/tex].
3. Distribute the negative sign:
Subtracting a polynomial is the same as adding its opposite (i.e., distributing the negative sign to each term inside the parentheses):
[tex]\[
-3x^3 + 5x^2 + 4x - 7 - 6x^3 + 2x - 3
\][/tex]
4. Combine like terms:
Group the terms with the same degree together and combine them.
[tex]\[
\begin{array}{rl}
= & (-3x^3 - 6x^3) + 5x^2 + (4x + 2x) + (-7 - 3) \\
= & -9x^3 + 5x^2 + 6x - 10
\end{array}
\][/tex]
5. Final simplified form:
The result of the subtraction is:
[tex]\[
-9x^3 + 5x^2 + 6x - 10
\][/tex]
### Filling the drop-down menu:
[tex]\[
\begin{array}{l}
\left[\text{Step 1: Rewrite each polynomial}\right]\\
\left[\text{Step 2: Express the subtraction operation}\right] \\
\left[\text{Step 3: Distribute the negative sign}\right] \\
\left[\text{Step 4: Combine like terms}\right] \\
\left[\text{Final Step: Simplify to get the final result}\right] \\
\end{array}
\][/tex]
By following these steps, we arrive at the correct solution [tex]\(-9x^3 + 5x^2 + 6x - 10\)[/tex].
