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]
