Question

select the correct step and statement.
julian factored the expression $2x^4 + 2x^3 - x^2 - x$. his work is shown.
at which step did julian make his first mistake, and which statement describes the mistake?

	$2x^4 + 2x^3 - x^2 - x$
step 1	$= x(2x^3 + 2x^2 - x - 1)$

| step 2 | $= x\left2x^2(x + 1) - 1(x - 1)\
ight$ |

step 3

$= x(2x^2 - 1)(x + 1)(x - 1)$

statement 1	julian should have factored $(2x^2 - 1)$ as a difference of squares.
statement 2	julian incorrectly applied the distributive property when factoring out -1.
statement 3	julian should have factored $2x$ from all terms instead of $x$.
statement 4	julian incorrectly factored $2x^2$ from the first group of terms.

Explanation:

Step 1: Analyze Step 1

Julian factored out \( x \) from \( 2x^4 + 2x^3 - x^2 - x \), getting \( x(2x^3 + 2x^2 - x - 1) \). This is correct as \( x \) is a common factor.

Step 2: Analyze Step 2

In the second group \( -x - 1 \), when factoring out \( -1 \), it should be \( -1(x + 1) \), but Julian wrote \( -1(x - 1) \). So he made a mistake in applying the distributive property here.

Step 3: Analyze Step 3 (Optional, since mistake is in Step 2)

Due to the mistake in Step 2, Step 3 is also incorrect, but we focus on the first mistake.

Answer:

Step 2, Statement 2 (Julian incorrectly applied the distributive property when factoring out -1)