Answer :
To determine if there is any error in the given steps, let's examine each transformation step-by-step:
### Line (1)
[tex]\[ 4x^3 + 2x^2 - 6x + 3 \][/tex]
### Line (2)
[tex]\[ = 2x^2(2x + 1) - 3(2x + 1) \][/tex]
To verify the correctness of the step from Line (1) to Line (2), we need to expand and simplify the expression in Line (2).
Expanding Line (2):
[tex]\[ 2x^2(2x + 1) - 3(2x + 1) \][/tex]
Distribute each term inside the parentheses:
[tex]\[ (2x^2 \cdot 2x) + (2x^2 \cdot 1) - (3 \cdot 2x) - (3 \cdot 1) \][/tex]
[tex]\[ = 4x^3 + 2x^2 - 6x - 3 \][/tex]
Now compare this with Line (1):
[tex]\[ 4x^3 + 2x^2 - 6x + 3 \][/tex]
[tex]\[ 4x^3 + 2x^2 - 6x - 3 \][/tex]
It is clear that there is a discrepancy. The constants in Line (1) and the expanded Line (2) differ by a sign.
- Line (1) has \(+ 3\).
- Expanded Line (2) has \(- 3\).
Thus, there is an error in the step from Line (1) to Line (2).
### Summary
Given our detailed check, we see that an error occurred from Line (1) to Line (2). Therefore, the correct statement is:
- The error occurred from line (1) to line (2).
There are no needs to verify subsequent lines as the error has already been identified in the initial transformation from Line (1) to Line (2).
### Line (1)
[tex]\[ 4x^3 + 2x^2 - 6x + 3 \][/tex]
### Line (2)
[tex]\[ = 2x^2(2x + 1) - 3(2x + 1) \][/tex]
To verify the correctness of the step from Line (1) to Line (2), we need to expand and simplify the expression in Line (2).
Expanding Line (2):
[tex]\[ 2x^2(2x + 1) - 3(2x + 1) \][/tex]
Distribute each term inside the parentheses:
[tex]\[ (2x^2 \cdot 2x) + (2x^2 \cdot 1) - (3 \cdot 2x) - (3 \cdot 1) \][/tex]
[tex]\[ = 4x^3 + 2x^2 - 6x - 3 \][/tex]
Now compare this with Line (1):
[tex]\[ 4x^3 + 2x^2 - 6x + 3 \][/tex]
[tex]\[ 4x^3 + 2x^2 - 6x - 3 \][/tex]
It is clear that there is a discrepancy. The constants in Line (1) and the expanded Line (2) differ by a sign.
- Line (1) has \(+ 3\).
- Expanded Line (2) has \(- 3\).
Thus, there is an error in the step from Line (1) to Line (2).
### Summary
Given our detailed check, we see that an error occurred from Line (1) to Line (2). Therefore, the correct statement is:
- The error occurred from line (1) to line (2).
There are no needs to verify subsequent lines as the error has already been identified in the initial transformation from Line (1) to Line (2).
