Select the correct step and statement.

Julian factored the expression [tex]2x^4 + 2x^3 - x^2 - x[/tex]. His work is shown. At which step did Julian make his first mistake, and which statement describes the mistake?

\begin{tabular}{|l|c|}
\hline
& [tex]2x^4 + 2x^3 - x^2 - x[/tex] \\
\hline
Step 1 & [tex]= x(2x^3 + 2x^2 - x - 1)[/tex] \\
\hline
Step 2 & [tex]= x[2x^2(x + 1) - 1(x - 1)][/tex] \\
\hline
Step 3 & [tex]= x(2x^2 - 1)(x + 1)(x - 1)[/tex] \\
\hline
\end{tabular}

\begin{tabular}{|l|l|}
\hline
Statement 1 & Julian should have factored [tex](2x^2 - 1)[/tex] as a difference of squares. \\
\hline
Statement 2 & Julian incorrectly applied the distributive property when factoring out [tex]-1[/tex]. \\
\hline
Statement 3 & Julian should have factored [tex]2x[/tex] from all terms instead of [tex]x[/tex]. \\
\hline
Statement 4 & (Empty - No statement provided) \\
\hline
\end{tabular}



Answer :

To determine at which step Julian made his first mistake and to identify the correct statement describing this mistake, we will carefully review each step of the factorization process:

1. Step 1:
[tex]\[ 2x^4 + 2x^3 - x^2 - x = x(2x^3 + 2x^2 - x - 1) \][/tex]
Factoring out [tex]\(x\)[/tex] from each term correctly, this step is correct.

2. Step 2:
[tex]\[ x(2x^3 + 2x^2 - x - 1) = x\left[2x^2(x + 1) - 1(x - 1)\right] \][/tex]
In this step, Julian factored [tex]\(2x^2\)[/tex] from the first two terms and [tex]\(-1\)[/tex] from the last two terms. Here he made a mistake. The correct grouping should be:
[tex]\[ x[(2x^3 + 2x^2) - (x^2 + x)] = x[2x^2(x + 1) - x(x + 1)] \][/tex]
Further factoring gives:
[tex]\[ = x[(2x^2 - x)(x + 1)] \][/tex]
So, Julian incorrectly applied the distributive property when he factored [tex]\(-1\)[/tex].

3. Statement Analysis:
- Statement 1: This is incorrect because Julian’s error wasn’t about factoring [tex]\((2x^2 - 1)\)[/tex] as a difference of squares.
- Statement 2: This is correct, as Julian incorrectly applied the distributive property when factoring [tex]\(-1\)[/tex].
- Statement 3: This is incorrect because the initial step of factoring out [tex]\(x\)[/tex] was correct.
- Statement 4: Not applicable as it is empty.

Thus, Julian made his first mistake in Step 2, and Statement 2 accurately describes this mistake.

Therefore, the correct selection is:
[tex]\[ \text{Step with mistake: Step 2} \][/tex]
[tex]\[ \text{Statement describing the mistake: Julian incorrectly applied the distributive property when factoring out -1.} \][/tex]